Can't comment specifically on exercise equipment, but...
There are a number of threads I think in this forum and BA that suggest the T3-T4 transfer (or vice versa) can be done just as quickly landside as airside assuming the immigration queue isn't too long. So you could check out the length of the queue, clear immigration, use the AA arrivals lounge and then take the Heathrow Express over to T4 and use the BA Terraces lounge before the flight to BLR.
People would probably say though to just go over and use the BA lounge in T4, because with a 4 hour layover, you'd have a good shot at getting a treatment at the Molton Brown spa. But again, I can't comment on where there might be an exercise facility.