Trouble is, the door is an original door built into the alcove at the
side of the fireplace so I'd rather leave it alone as much as possible.

The remote sender is a good idea, but I actually have a Freeview box,
and a surround-sound unit in there too - so I'd need 2 remote sender
kits. A bit expensive really.

Leaving the door ajar works, but, not very pretty!