What are you envisaging when you say "I can get a reward flight to Miami whilst keeping my original booking"? If your plan is to leave your BA Holiday booking unchanged, and to simply not turn up for the LHR-BOS flight, this will not work. Your return JFK-LHR flight will be cancelled due to you being a no-show on the outward flight for that booking.

I would suggest you either pay to change your existing BAH booking to fly out to MIA (though this is likely to be expensive if this is at short notice), or leave it as it is and lose the deposit. If you choose the latter, redemption availability coming back from New York is generally decent so you shouldn't struggle to find something that works for you.

Another option is to fly the BAH as booked, and book a separate BOS-MIA flight (which could be done with Avios if you fly with American Airlines). That may be the cheapest option of all and would retain as many of your Avios as possible. NB you have no protection in the event of a missed connection on separate bookings, so I would suggest allowing plenty of time between flights or alternatively staying overnight.

Booking a BOS-MIA redemption through Qatar Airways would reduce the cost from 18k / 40k Avios (economy / domestic "First" if booked via BA) to 14.5k / 34.5k respectively. It would also give you the option of redeeming on jetBlue for 13k Avios (NB they only have economy and aren't a Oneworld member) which BA doesn't offer you. You can transfer Avios for free between your BA and Qatar accounts - you need to link them first.

NB you can 'boost' your Avios at 0.92p each via the BA site, and you can get a monthly subscription earning you Avios for approx. 1p each through BA, as well as Finnair and Qatar Airways. So I wouldn't worry too much about using up Avios; if anything, I would suggest holding onto as low a balance as possible because there is the suggestion that redemptions will become dynamically priced in the future, which would essentially trash their value (that's exactly what happened with Virgin Atlantic's scheme).