Phil,
While I agree that unplanned expenditures should be controlled, and you must set limits, the situation here is not the Principal's, but the PTO representative (Exec board, committee head etc) that has fiduciary responsbility to the PTO. Thus, giving an independent party like the Principal does not allow initiation of the special expense by the Principal, just confirmation.
Of course, just about any rule can be abused, which is why verbal confirmations, emails etc can help when enough officers can not be there. And the smaller the limits can be set, the less chance of abuse of funds. Hopefully, you can strike a happy median.
No one says that one can't have different "emergency fund" amounts for different situations. E.g. see a sale of items for the Holiday Shop, and needs to be picked up might be $25 max. On the other hand, the Spring Dance might need more latitude with expenditures, so perhaps $150. These could be hard coded in the bylaws, or allowed to be set by the Exec Board or similar, or a combination of both (I would always set some cap in the bylaws, so there was no way to just change it at a whim)
In any case, there should always be reporting of all expenditures both in reports and to the membership, even including unreimbursed expenditures or donations. This is important to better plan future expenditures, where you can't always plan on someone donating or eating the cost.
A final review of money situations that have occured should be performed to help eliminate their cause in the future.