Assuming this is for your trip in June 2023, the Elizabeth Line should be running by then all the way through central London direct from Heathrow, giving you more options. (It currently involves a change at Paddington, but the through trains are due to start this November).

So, your most comfortable option may be to take the EL direct from Heathrow to Whitechapel, and then step-free change there for 2 stops on the District Line to Tower Hill. The Elizabeth Line is very comfortable and will feel quicker than the Piccadilly Line, albeit is a bit more expensive. You would definitely get a seat from Heathrow.

(Alternatively, you could change from EL at Farringdon or Liverpool St but the latter is probably a longer walk. Citymapper recommends the Whitechapel change.)

If you prefer to take the Tube, you would similarly get a seat at Heathrow and the Piccadilly Line bit would be fine. Changing at Barons Court is then easy, but trains could be fuller from there. If you were to ask people they would happily move for you, if not offering pro-actively.