Just to close this out...we did end up transferring directly to CPH from the ship, but left later than expected and didn't get to the airport until around noon. So we decided not to return to town and instead chatted for several hours with another couple we'd met several times on the ship and who also happened to have lounge access with their ticketing.

Bag drop was uneventful and we were able to check the bags all the way through to LAX. Bit of a ding for the (obviously now outsourced?) EC Customer Service chap I talked to a week before the flight though, who told me that we would NOT be able to check our bags all the way through. I had a feeling that was wrong, so we still packed our carry-ons accordingly for the Sofitel overnight. But I could really have done without the uncertainty, if I had to be honest.

Flight from CPH ended up being 30 minutes late departing (some small problem with the brakes), which pretty much blew the already slim chance we had of getting into town soon enough for some theater that evening (we were trying for 'Jersey Boys'). But we did have a very nice kebab dinner (don't knock it...that's what we wanted).

Sofitel was very nice, and I wouldn't have any qualms about staying there again and taking the tube into London.