yes you can stay airside.
T3-T5 airside really isn't as bad a process to go through as some people make it out to be. Just follow the purple signs and security at T5 is the same as you would have done on departing MIA for example.
You can do it airside if you want - clear the UK border at T3 then use the tube / train (both free) to get to T5 and go through security etc