If you did want to take the train, Feltham could be an easier option. Southern train to Clapham Junction, walk from platform 12 to platform 6 (yes, a pain with luggage again) and then SWT to Feltham. Bus or cab from Feltham to Heathrow. National Rail reckon an hour, but in practice, you can do it in less with a quick change at CJ.

However, this is still quite a hassle.