We did something similar earlier this year; arrived on BA (in First on Avios) and connected to AA (albeit the next day). I think we were through immigration about 20 minutes after the plane landed (to be fair we're US citizens but they had a ton of people working and it didn't look that bad). Took another 15-20 minutes to get our bags. We took ground transport to a hotel to overnight because we didn't want to rush to make our AA flight to the west coast but the next morning check-in took all of five minutes with AA using their priority lane and we were through security in five minutes with pre-check. All that to say, I think that having 4:15 is more than enough. The Admiral's Club is nothing special but it's still got some small snacks and a bar and you'll get two drink coupons for alcohol or smoothies or bottled water.. If you want to do what we did, there is a Marriott in the terminal that you can stay at and then leave first thing the next morning.