Why doesn’t the House just buy the St. Ermin’s Hotel, then they can stay there if they need? IIRC it has a passage through to the Palace of Westminster (for those that don’t know, that’s the name of the building that houses Parliament), and it even used to have a bell in the lobby that rings when there’s a vote in the House, so that MPs can make their way through to Parliament to vote.

Then there wouldn’t be any need for a second home allowance, or any kind of overnight allowance for that matter either.