I hope FT has given you confidence that the Visa is now not problem if you go with an agent.
No personal experience of admitting a second person into a room booked for 1, but the IC room rates are per room not per person, so there is no cost difference. I would be absolutely gobsmacked if they even blinked at booking the two of you into the room and registering you. Some of the smaller hotels might baulk, but a major chain like IC isn't likely to mess about. I never use the hotel for the invitation, I only book the hotel once I have a Visa.
Booking a room, using it for an invitation and then cancelling it is not something I would consider. YMMV.
Jealous of your IC experience. Seems to be well priced right now. Stayed in The National next door but not the IC.

I would just present your card on check in with significant other. You might not get the upgrade benefits you would expect pre/on check-in but IMO status is worth flashing in Russia.

To add BA relevance. Two trips to Moscow in BA F have been my only CCR experiences