I haven't stayed in either hotel, but did just get back from Florence. From mapping both, it appears the Bernini is definitely in the middle of things (in a good way) while the the Lucchesi appears to be overlooking the Arno near Santa Croce.--probably no more than a 10 minute walk from one to the other.

Interestingly enough, the Bernini is rated the 53rd most popular hotel on Tripadvisor, while the Lucchesi is the 54th. Both Bernini and Lucchesi have good writeups.

Hope this helps. If it was me, I'd ask for some compensation, but I think I might rather enjoy being on the Arno (possibly quieter?).