Many thanx for your responses.
If you can find a route which gets you into Miami by 11:00, then AA might agree to it, but if you simply prefer to spend the night in city X rather than Miami, then they certainly aren't obligated to accommodate that.
The earliest LHR-MIA flight AA57 arrives at 2.25pm. No route on same day that gets me in to PUJ via MIA. Should I accept AA's offer and leave a day earlier, are they obliged to cover overnight MIA hotel expenses?