Q:

What balance pitfalls result from this house rule modifying the Shield Master feat?

For an upcoming game, I'm weighing Shield Master as a feat selection. The closer I look at it (and recent rulings on it), however, the more lackluster I find it in comparison to other choices like Great Weapon Master, Polearm Master, Sharpshooter, and even Dual Wielder.
Shield Master's third benefit, the Evasion-like ability to avoid damage on a successful DEX save, is flavorful and useful. It embodies a classic narrative trope of shield-use, and it's likely to save significant HP that would otherwise be lost to breath weapons, fireball and similar spells, etc.
The feat's two other benefits are more dubious.
The first benefit, granting a bonus-action shove if one takes the Attack action, has been the subject of rule-reversals restricting its application to after all of one's attacks are completed. I find that restriction dissatisfying both mechanically and narratively. Mechanically, it means a character using the bonus-action shove will rarely benefit from doing so, because her target can simply move when its turn comes around, before she has a chance to act again. And narratively, forcing a melee combatant into an "attack-shove" (or, for those with Extra Attack, potentially "attack-attack-attack-attack-shove") routine feels arbitrary and unnatural. Imagine a tale of heroic adventure in which a character described as "master" of shield technique did not, could not, ever lead with her shield. For that matter, why is it so difficult, even for a character with Shield Master, to effectively use a shield-bash as a damage-dealing option in melee? Overall, compared to the highly effective and directly beneficial bonus-action attacks granted by Great Weapon Master and Polearm Master, this part of Shield Master is underwhelming.
The second benefit, adding the shield's defensive bonus to DEX saves against single-target effects, sounds useful in theory but turns out to be disappointingly narrow and situational in application. It's not unrealistic to think a character with Shield Master could make it through an entire campaign and never use this benefit. It's hardly a benefit at all.
Therefore, I'm considering a house rule modifying Shield Master, by adding the following two bullets:

You are proficient with shields as improvised weapons.
When you are wielding a shield as an improvised weapon, you can use two-weapon fighting even when the one-handed melee weapon you are wielding in your other hand isn't light.

This modification seeks to put Shield Master roughly on par with Polearm Master and Dual Wielder. A character with Dual Wielder can wield, e.g., two longswords with a d8 damage die, or a d8 longsword and a d4 whip with reach, and still gain a +1 to AC. A character using Polearm Master won't get a shield's AC bonus, and will only have a d4 damage die for her offhand (i.e., "opposite end") attack, but adds her ability modifier to the offhand damage, is likely to enjoy a polearm's d10 damage die for her primary attacks, and has reach on all her attacks. By comparison, with this modification, a Shield Master character can enjoy the +2 bonus to AC from her shield and still utilize two-weapon fighting, albeit with just an unmodified d4 damage die from the shield. Or, thanks to the proficiency piece, she can viably lead with her shield, treating the shield as her primary weapon and using her normal weapon for a bonus-action attack with unmodified damage. She could even lead with a shove using her Attack action and still get a bonus-action attack via two-weapon fighting -- just as she could with Polearm Master or Dual Wielder.
Nevertheless, it's possible I'm missing something. Are there foreseeable balance problems here? What are they?

A:

It's probably overpowered
The thing with shields is that they add a big boost to AC at the expense of increased damage. Usually the shield prevents you from holding two-handed weapons which have the bigger damage die - or prevent you from holding a second weapon for two weapon fighting.
Turning that into another attack option skews this to something everyone is going to want to have.
+2 AC (or more with magic shields) and I can still attack with that arm? AND I get the other shield master attributes? YES, PLEASE!

A:

Your modification actually changes two things
I'd be careful about changing more than one feature at a time.  By also changing two weapon fighting, you are not just changing this feat but also a core game feature.  By removing the light restriction, you are offering the benefit of two feats for the price of one.  (Or one and a half, since your shield's damage is limited).  
Your modifications focuses more on dealing damage.  Flavor wise, I like the idea that the shield now becomes a weapon, but "fiddly" wise I'd offer a caution.   If you add the attack feature, I'd suggest you leave two weapon fighting alone.  The Dual Wielder feat is the one that lifts the "light" requirement from two weapon fighting.    
The numbers comparisons are going to be very table dependent.  

Who is in your party and how do you all fight?  
If you have another front-liner in your party, losing out on them getting an attack, or multiple attacks, with advantage once you knock an opponent prone may hurt your party's overall DPR.  Your bonus action based improvised weapon attack hardly substitutes for your ally taking two attacks with advantage, or, if a Monk, maybe four attacks with flurry of blows.  
If you are the sole front liner, this move allows you to increase your DPR without sacrificing armor protection, and allows you to not be required to use a light weapon for two weapon fighting.  
A word of caution, on balance: with the bonus attack from the shield as an improvised weapon, on a Paladin of level 5 or higher, you have three chances per round, not two, to trigger the Paladin's Divine SMite.  This is bursty damage limited by spell slot availability, but this increased DPR opportunity swings farther toward "not balanced" for this particular class.  At level 4 or lower it's two attacks, not one, that can trigger the smite."  

My at-table experience does not concur with your linked "down side" to the defensive features, nor to the "overpowered" charges that led to Crawford's backpedal.  
Having played SM both ways ... with a Fighter / Champion1 ...

Nothing significant happens in either direction for balance.  Our
table, when I selected shield master with a feat, had ruled with the
original SA case that the ordering of that bonus action was not
limited to before or after.  The DM changed it when Crawford
backpedalled.  
On the side of "flexibility" the limitation that the shield-bash has
to come after the attacks means that the player has fewer options
for what to do tactically, particularly when there are multiple
attacks available to the Champion.  
(Levels 10-141)  
What our group figured out was that we could "make the most of the
change" to the more limited case by trying to knock things prone so
that another attacker (usually our Barbarian) could follow up with
advantages on the next attack (without using Reckless Attack).  That
varied in effectiveness, but since he used Great Weapon Master with a two handed weapon, it made for some bursty rounds.  I also took the chance to push things off of cliffs, and in one case pushed a troll into a pool of lava.  (Heh, one shot a troll FTW!)  
The assertions of "overpowered" are groundless, based on what I saw
in play at the table.
One of the nicer features of Shield Master feat is that combination
of those two defensive features.  It is a mistake to assume that
those with a Shield Master feat will not also have a Dexterity with
a modifier +1 or higher.  In tier two or three, a +1 Shield would not be unusual to find on a sword and board fighter.  {Paid Political Announcement: Fighters do a bit better with better initiative: don't neglect Dexterity in your Fighter builds}.  I don't see it getting as much discussion even though you allude to it.  It applies when attacked, so you don't control its activation as much, but when it works it saves a boat load of incoming damage.  

If you aren’t incapacitated, you can add your shield’s AC bonus to any Dexterity saving throw you make against a spell or other harmful effect that targets only you.  
If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you can use your reaction to take no damage if you succeed on the saving throw, interposing your shield between yourself and the source of the effect.

The breath weapons of a number of creatures, enemy fire balls,
lightning bolts, and more all target the dex saving throw.  That
feature alone went a long way to keeping my Champion in a lot of
fights.  That is a reaction well expended when a pack of hellhounds
are all breathing at you as you block the doorway while your party
re-positions ...

My recommendations

If you are going to change this, change one feature at a time.  Playtest it first with two weapon fighting unchanged.  
Take Shield Master as is, and use your teamwork and wits to get the most out of its existing features.  

A:

It reduces player choice by diminishing the attractiveness of other options
As NautArch points out, making the changes you specify essentially allows a player to have her cake and eat it, too. It creates a clearly optimal character build choice that diminishes the choices for players.
That is, Even if you believe that the benefit is minimal, melee characters will probably feel obligated to take the feat or admit that they are choosing to be underpowered. Maybe that's okay with you, but it's a consideration to be aware of.
For what it's worth, one of the big advantages of Shield Master is how it synergizes with grapple-based builds: a character can grapple a target with their attack and then shove it prone as a bonus action, rendering it unable to standup until it breaks the grapple. Yes, the individual character is giving up an attack roll but each of the other melee characters get to roll an additional die to hit.
A character with your modified feat wouldn't have to weigh the tactical merits of grappling and versus making attacks for higher than 1+Str damage. They could do both.

