[timtlm]

It appears that if I start streaming after the phone becomes fully charged, it will not drain the battery. I can tell because it doesn't get as hot.

So, I don't see why it can't do it when the battery is not fully charged to begin with. It just needs to dedicate most of the power to operating the phone and whatever is left to charge the battery. Instead it must be letting the battery operate the phone, and then lets the adapter charge the battery as long as it doesn't get too hot. Once it gets too hot, it stops charging. Seems like this could be fixed with a software update.