There shouldn't be any conflict of interest, the two bodies are entirely separate, plenty of PTA Treasurers are also governors and as long as they understand the distinction there should be no issue. There may be a rare instance where they would have to declare a conflict of interest at a Governors Finance meeting, but it would be a very rare occasion. PTA funds would not usually form part of Governor's meetings.
However, it sounds as if something else is going on here.
The last committee were basically forced out by this CoG who persuaded the Head to go along with a virtual coup.
Not understanding this at all. The Head nor CoG were part of the committee at the time? How did they force them out? If the committee just got fed up with interference they should have quit, then an EGM be called and the members (mostly parents) vote in a new committee.
A new (impartial) Treasurer was voted in at the AGM but this person has now seen what a clique the whole thing is and wants to step down because no one will listen to them. Lo and behold, now the CoG wants to do the job to "help out".
Ask if any other parents want to do the job? Encourage the other committee members to be welcoming to the new addition. If nobody else steps forward apart from the CoG then there may not be any other option as you must have a Treasurer. However, that doesn't mean that they can make the decisions on where the money goes. It isn't up to the Treasurer, it's the decision of the committee as a whole how the money is spent. That decision needs to be made in line with whatever is outlined in the constitution. Ideally it's a good idea to talk to the school about what they would like additional funds for, and some schools engage their wider membership with ideas, but the final decision lies with the committee as Trustees.
FWIW the CoG is an accountant so fully qualified!
There are no qualifications required to be a PTA Treasurer. Many are accountants, purely as they are often the people who volunteer for this unpopular role! (I've been trying to offload my job as Treasurer for several years to no avail, this year it's looking like I finally might have a successor! yay! unsurprisingly they are an accountant). If you can keep track of money and know how to use excel you are good to go! You could volunteer as the new Treasurer.