The two hotels you mentioned are very close, but the Bernini is definitely in a very slightly better location. It is far more classy than the Palazzo Borghese, which is a bit "pseudo," whereas the Bernini just oozes class. What I like about the Bernini is that is perfect for collapsing back into when you emerge, hot, exhausted and brain dead, from touring the Uffizi. It is right next to the exit and perfect for having a cold shower, cold drink and just relaxing. The hotel staff are very highly trained with a genuine Florentine twist. I say cancel your Borghese booking and go for the Bernini!
I also think you should stay the night in Pisa and make your way to Florence the next day. Get Pisa out of the way and then utilise the rest of your time in Florence. Trains and buses go from Florence station right to Pisa station.
Edited: 8:06 am, March 08, 2013