With the timings the OP has provided, I would focus on transferring directly from T5 to T2 upon arrival, via the airside buses (look for the purple Flight Connections signs, it is very easy), and then once in T2 use your Priority Pass to enjoy the Plaza Premium lounge, which I haven't visited but is well reviewed and of a high standard I understand.
With a 3.5 hour connection I wouldn't recommend going through the rigmarole of clearing immigration, visting the landside Arrivals lounge at T5, then travelling via Tube or Heathrow Express to T2 (then having to clear security at T2 etc) - personally I think there is not enough time for you to do this without quite a bit of rushing around and trying to navigate unfamiliar airport terminals.