For fixed hypotenuse, can the number of primitive Pythagorean triples exceed the number of non-primitive ones? For the equation,
$$a^2+b^2=c^2$$
if $c$ is fixed and the number of natural solutions for $a, b$ is greater than $1$, then can the number of primitive solutions (solutions in which $a, b, c$ are coprime) exceed the number of non-primitive ones?
After testing a large number of cases I believe that the number of non-primitive solutions will always exceed the number of primitive ones although I have no proof for this.
If false, what is the smallest counter-example for $c$?

Given Euclid's formula $\quad A=m^2-n^2\quad B=2mn\quad C=m^2+n^2\quad $you can find Pythagorean triples given only $C$, if they exist, by solving the $C$-function for $n$ and testing for a defined range of $m$-values to see which, if any, yield integers.
$$C=m^2+n^2\implies n=\sqrt{C-m^2}\quad\text{where}\quad \biggl\lceil\sqrt{\frac{C}{2}}\biggr\rceil \le m < \lfloor\sqrt{C}\rfloor$$
For example, I believe this is the $\textbf{smallest counter-example}$ you were looking for:
$\quad C=65\implies n=\sqrt{65-m^2}\quad \text{where}\quad \biggl\lceil\sqrt{\frac{65}{2}}\biggr\rceil=6\le m \le \lfloor\sqrt{65}\rfloor=8 $
Testing,  we find $$F(65,6)=\sqrt{65-36}=\sqrt{29}\notin\mathbb{N}\\F(65,7)=\sqrt{65-49}=\sqrt{16}=4\quad\quad\quad F(65,8)=\sqrt{65-64}=\sqrt{1}=1\quad$$ which yields
$$f(7,4)=(33,56,65)\quad\text{and}\quad f(8,1)=(63,16,65)$$
For $C=1105$, we have $24\le m\le 33$ and
$$f(24,23)=(47,1104,1105)\qquad\qquad f(31,12)=(817,744,1105)\qquad\qquad 
f(32,9)=(943,576,1105)\qquad\qquad 
f(33,4)=(1073,264,1105)\qquad $$
For many $C$-values, non-primitives out-number primitives but $\textbf{all of these triples are primitive}$ and many more are easy to find.
Euclid's formula generates primitives, doubles, and square multiples of primitives but not, for example $(9,12,15)$ or $(15,20,25)$ which are $3\&5$ times multiples of $(3,4,5)$, respectively. If this formula does not find a triple for a given value of $C$, try any of the factors of $C$ that take the form $4x+1, x\in\mathbb{N}$.
