Hmmm... Coming the other way (AA connecting to BA) there is a special airside shuttle bus from T3 to T5 just for BA passengers (you have to show your onward BA boarding card to use it)... I doubt it works the other way as you wouldn't go through the US immigration controls.

I would allow plenty of time for getting through security at ORD... I've stood there for well over an hour before.

With regards to which lounge - I think that you are sensible to get to T3 ASAP. As you are a GC holder you should be able to access the Flagship Lounge at ORD... This is better than the Admirals Club lounge (but nothing like as nice as Galleries First at LHR)... I remember it was a bit of a walk from my gate, but hopefully you would be closer.