The Heat might be able to solve their cap problem in an unexpected and ironic way..
Their main target now for the Heat is Deng, who happens to be on Cleveland's books right now (which is incidentally where Lebron is heading in case you missed it). The idea (if they agree with him on a deal) is to not sign him as a free agent (for cap space) which would limit their flexibility greatly by depriving them the right to use all of their exceptions (including bird rights to re-sign their current players), but to make a trade for him with Cleveland (Deng for Lebron, which is a bit unfair I know but it's ok), and that will not only help them retain all the exceptions they have including the MLE, but will also enable them to offer competitive money to all of their players from last year.
The only catch is, Dan Gilbert and Cleveland helping the Heat? Not sure about that... However, if Lebron is going to do the same thing he did when he came to Miami (demand the fifth year on his contract), then he can easily facilitate this trade (in the same way he got Cleveland all those draft picks from Miami in 2010). Will he do it?
If Lebron doesn't want to cooperate, then the Heat might need to give away even more draft picks to Cleveland to make it possible, which would pretty much ensures that every single draft pick the Heat had in this decade is going to Cleveland.
Of course it's still someway away (the Heat are still negotiating with Deng over salary), but I thought it's an interesting scenario just to think about it. And who knows, it could happen.