You almost certainly will have to pay the underage fee for the entire rental. The computer systems are just set up that way--they look at the age of the driver on the start date of the contract and the remaining length of rental is irrelevant. The way around it would be, as was already suggested, to break the rental up into two sections, but you mentioned that would not be convenient for you. It's possible you might be able to get a manager to issue a manual credit for the days after she turns 25, though I would not count on that.

The wiki in this thread contains the latest information we on FT collectively know about avoiding underage fees: The Definitive Young Renters Thread: 18-20, 21-24. Of note, it does briefly mention that the Visa cardholders AWD, B204400, waives the fee as an unpublished (and perhaps unintentional) benefit. You mentioned booking through BA; I would see if you can rebook the rental there to include that AWD code. If not, try booking directly on avis.co.uk so you can still get the European-based all-inclusive rates (so you don't need to purchase coverage) but can apply the AWD there. Can't guarantee the fee will be waived, but it's probably worth trying.

As for the IDP, it basically serves as a translation of your license. If your license is readable by someone who only speaks English, it generally won't be required. If your license contains only Chinese characters or Arabic script or something, the rental agent will probably be lost without the IDP. While it is possible that the fine print with some companies does require an IDP, you're likely OK without it. (That said, I do make it standard practice to always have one when traveling overseas, because it would royally suck to get 5,000 miles away and not be able to drive because of it. But AAA only charges $10/year for an IDP here.) Best way to find out the official policy is to call the rental office directly; online T&C often don't mention IDP requirements, IME.