You can't book the work trip without the danger of AA's detecting the overlapping flights and cancelling one. You can try not including your FF number, but sometimes corporate travel systems include FF numbers automatically. You could fly on a different airline, but it sounds like you want to earn AA miles, so that would defeat the purpose.
You could cancel all or part of your return to FLL, but what would you do if your business trip were to be changed or cancelled at the last minute? You would be stuck without a ticket home.
I would keep the trips separate. Yes, it will be exhausting to cab it to MIA after your arrival at FLL, but sometimes mileage runs are like that.
