Sounds like an awful situation, unfortunately I wish you had posted here first as we could have sent you down a much better path:
1) Since your ticket was booked through AA, you need to always call AA for all changes (even though it was a BA flight). Calling BA will only confuse/compound the problem.
2) Since it was a schedule change you are free to ask for any new flight you want, in business class, regardless of award availability. This is where asking here first would have really helped.
3) Since your new flights had over 24 hours in London, I'm guess the AA agent had to split the award into 2 awards BUD-LHR + LHR-BNA in order to reissue the ticket, which resulted in the extra miles and extra $$ taxes collected (UK has very high departure taxes which you would not have paid if you were only on a connection).
Unfortunately now, about the only thing you can do is write to AA customer relations online at aa.com and request a refund of the $725 and 22k miles. As mentioned you need to keep your story much shorter when you write in, just the key details.."There was a schedule change, I was forced to pay an extra $725 + 22k miles for new flights when other flights on the original date would have worked, etc. etc."...