How to change serial number on a Mac Pro 5,1 (2010)?
I may have misunderstood your question, but it's not clear what you mean by:
 my account works fine on other machines with different serials, yet even on this Mac Pro with a completely fresh install, it absolutely will not work.

I'm not aware of any Apple ID account services that interrogate serial numbers. In fact, Apple instructs software developers the following caveat relating to serial numbers:


  It is possible for a system to lose its serial number so that it will
  no longer appear either in System Profiler or the I/O Registry.
  Repairing a system by swapping hardware components is one reason this
  can happen. Apple does not document the specific details of how a
  machine can lose its serial number. Once the serial number has been
  lost there is no means to restore it to the machine.


Source: Apple Technical Note TN1103

The point I am trying to make is that Mac computers can lose their serial number, or the serial number can change (e.g. through a replacement logic board), and because of this serial numbers are only used by Apple as one way of identifying/tracking computers for Warranty purposes (e.g. relating to Apple Care, recall programs, etc).

This being the case, it's not clear what isn't working on your Mac Pro in terms of your account. If you could clarify, perhaps the community may be able to address what isn't working in a different way? Please don't take this as me being dismissive - I am merely trying to find a way forward for me or someone else to help you. 

Missing serial numbers

I have seen a number of cases where users thought that a missing serial number was the cause of problems (such as the Messages app being unable to connect) but the real issue was something else. Evenso, in these cases the scenario was a missing serial number not an invalid one. (I should add that I have personally seen Macs with missing serial numbers function normally in all respects.)

Valid serial numbers

In the previously mentioned Tech note, Apple also states Developers should not make any assumptions about the format of the serial number such as its length or what characters it may contain.

This being the case, I'm not sure how you could generate a valid serial number for your Mac Pro. 

If, on the other hand, your serial number was missing altogether, there is a tool that allows you to re-enter a serial number, but this tool only works if no serial number is present - it doesn't allow you to edit an existing record, which you do have.

Proving your hypothesis

By the way, if you are correct about what is going on in your case, the only way you will prove it to Apple is to take these steps:


use your Apple ID on your Mac Pro and demonstrate what isn't working
use your Apple ID on another Mac Pro of the same model/configuration and show that everything is working fine
swap the logic boards between the two
use each computer in turn
if everything works fine on your machine with the other logic board, and no longer works fine on the other machine with your logic board, you've proven your case