Except the Ren isn't that near any tube stop. So you'd either have to ride a bus into Heathrow from the Bath road (free), pay for the slow and infrequent Hotel Hoppa, or take a longer bus ride to some other tube station, like Hounslow West.

Added to the 45 minutes or so it takes to get from LHR to the center of London on the Picadilly line, that sounds like an unpleasant trip to make twice each day, every day.

Why not stay at Regents Park the entire time and taxi it to Paddington for the Heathrow Express on your last day, unless you are leaving really early before it starts? It might be just about as fast!