[NAME] , yes, Xeon E5-2640 v3 should work.  But E5-2670 v3 or E5-2680 v3 would give you best value.  You can find used one for less than 100 USD on eBay server stores.  More cores and lower TDP (120w), it also will work with your existing heatsink. You are right about this workstation is picky about ECC memory.  Adding memory to your system will required a bit more work.  Using memory configuration chart for your system to find which will be the best configuration to add more capacities you needed with the least cost.  Getting the same clock, rank, will keep them running at optimal (2133 MHz).  In return, your upgrade will be problem free.
I am wondering if I can purchase a dual socket CPU (Xeon E5-2640 v3) and put it into my T5810 that currently has a single slot motherboard with a Xeon E5-1603 v3They are both the same socket type and I can Google and see some T5810s ship with a single 2640 but I'm trying to find something more official to confirm everything. Has anyone done this before or has a setup like this that can confirm? Also is this Precision tower as picky about the ECC memory that it takes like some of the Poweredge servers are or will any DRR4 ECC memory work?
Precision T5810 Using Dual Socket CPU with Single Slot Motherboard?