I've no doubt that you'll get lots of advice on the subject.... but I can't imagine this happening! PTO and school/district funding have to be separately accounted for. Schools being funded by tax dollars and, as a public entity are subject to GAAP and all that goes with it.
Your PTO is (or should be) a separate legal entity, funded entirely by donated funds. Your By-Laws should clear delineate independent audit requirements, etc.... They should also clearly articulate who can commit funds and signature requirements for checking, thresholds for dual signatures, etc....
I don't envy you having to tell a new Principal "no". But being "new".... agree to work with him/her to help educate him/her on how things have worked (hopefully well) in the past... and see if you can dig deep enough to find our what s/he really wants or is trying to accomplish.
I agree with JHB's analysis of different options, however, I feel pretty strongly that the right solution to the "transitory nature" and "principal/school will still get egg on his/her/its face" problems is good controls on the PTO's independent system.
The principal who wants to use PTO funds as private piggyback is, in my opinion, wrong. The principal who rightfully wants to avoid problems can do so by helping ensure that PTO puts smart money management practices in place. And the principal can actually play a key role in making sure those practices are continued with new officers.
I think there are serious involvement downsides when the principal runs things (the most motivated parents will be demotivated when they realize the principal runs the show), and those downsides outweigh any benefits of central control.
From what we've seen posted in the past on this Forum, it seems most PTO's do keep their finances separate.
However, we've also had some members write in to say their district's policy was that the school manage the account. It was still kept separate, and PTO still had control of how the funds were used, but the school basically acted as treasurer. With the many cases of embezzlement and funds mismanagement, it's not surprising some districts would want more control. Yes, the PTO is a separate legal entity. But, to a school, it's often a transitory group of volunteers where the management and financial controls can vary widely from year to year, depending on who's in charge.
We had an embezzlement issue a few years ago, and I can assure you - separate entity or not - the school and principal took a lot of flak. Technically, it was the PTO's money and the PTO's problem. But people naturally link the two and see the PTO as sort of an arm of the school. Like it or not, they end up sharing the blame if something goes wrong.
We've also heard from schools where the principal wants to treat the PTO account like his private piggy bank. I think we on the Forum are universally opposed to that.
So to answer your question (finally), my main concern would be that the PTO retain full authority over the use of the funds. Usually that means you also control the account. However, if we had that assurance, and our school district wanted to provide the bookkeeping function, I'd go for it. Unfortuntately, I suspect that's not what they are offering.
Our pTA is a seperate entity from the school. I would be leary of bulking it together with school funds also, you lose all control and to me thsat would be extremely uncomfortable.
In all the years I've been involved in PTA's and PTO's, I've never had this question come up. The new principal at our school wants us to run the PTO funds through the school's books. I've never heard of that being done and am against it. Am I right in thinking that PTA's/PTO's, since they are not part of the school entity, keep their funds completely separate from the school's? All the PTA's and PTO's I've been involved with have always had separate checking accounts that were not managed by the schools.