I have done it on many occasions, and have no problem with it at all (Admittedly hand luggage only)

The bus journey, on a normally very empty bus, is a maximum of 10 minutes if you're at the very end of bath road, and the walk from Heathrow Central to T3 is another maximum 10 minutes (allowing for walking really slowly dragging a big case).

Lots of options for an Accor Loyalty Member, which is why I highlighted it.

Obviously, if you value snob factor then it's a non-starter. If you value simple, free and quick transport connections then it's certainly a goer.