I wasn't even aware such a rule existed. My husband and I have checked in extra early many times, once almost 9 hours before the DFW-GRU flight. Who told you that the max is 4 hours? Sometimes the phone agents don't really understand airport procedure. Of course, there's always the possibility it's a new rule, it's been several months (at least) since I've checked in at the airport more than 4 hours in advance.

Although I've heard lots of horror stories about the AC at MIA, I think that might be your best bet. Best of luck.