This is just personal opinion but I have been to all the T3 and T5 lounges and I would not try to go airside in T3 in your situation. It depends on your priorities obviously, but if I'm connecting AA T3-> BA T5 with 4+ hours but not enough time to go into the city, I would likely go landside to the AA arrivals lounge (which is much less hectic and easier to get a shower than the airside lounges in T5), shower, eat, and then transfer to T5 landside, and spend any remaining time in the T5 lounge. I would be willing to do this (rather than going through flight connections, which is slightly simpler) because I put a high priority on showering when coming off a TATL flight.
I would not voluntarily go through security (which is an almost uniformly unpleasant experience at LHR) twice just to access the CX lounge, particularly first thing in the morning where my main priority is showering and getting a basic breakfast. Two trips through security plus the terminal transfer, all the walking, and leaving enough time to get to the gate for your onward flight, could easily eat up the majority of your connection time.