How would you get to Heathrow today if not by tube? And if not going by tube today, why can't you not go by tube tomorrow?

The Piccadilly line with luggage is not that unpleasant if you're at Earls Court before 1600. Just get on at the back of a carriage so that you aren't fighting the acceleration. If the trains are coming in e.g. 4m,6m,7m then the third train will not be crowded.

If you don't want to spend £40 then I'd prefer the tube with luggage over going to LHR twice and spending around £40 plus wasting 2 hours.