No easy answer to this one.

I have done the BA to AA connection at IAH before on two different itineraries though it has been a few years. You come out of IAH in terminal E, and then have to get to terminal A to the AA check in desk landside. IAH has a great train ("SkyWay") to connect the terminals, but it is airside. There is also one landside ("Subway"), it is old and not many people use it anymore other than a few employees. That said, it still works. So, it is very possible to collect your bags from terminal E, get on the Subway, and get to terminal A to recheck your bags.

It would be easier, as far as check-in goes, to fly United from IAH to MIA. United has a large presence in terminal E where you will come out of customs. You can re-check your bag with ease, and then take the SkyWay airside to your terminal.

Minimum connection time at IAH is 90 minutes. For an unprotected layover such as this, I would leave at least 2.5 hours, but that's my preference. Sometimes BA is late. Sometimes security lines are long. If you fly AA, they might be lenient if you miss your flight as they are both OneWorld, but no guarantee. If you have status with either AA or UA, that might also give you some flexibility.

The admirals club in terminal A is very basic. I don't think I would base my booking around access to that club. There are better restaurants all over that airport. If you do get access, it's a reasonable place to spend some time.

Final thought - keep your eye open for a deal directly to Miami. BA has a few flights daily that go there direct and you can skip the hassle. Hope this helps.