I understand that a furnishing bag, similar to the craft bag, won't work because of the sheer number of furniture items there are.
What would work, I think, is a furniture warehouse.
Someplace down by the docks
When you open the door and enter the building you are ported to another server and there is a menu. That is it. The only thing one can do is move furniture between the warehouse and inventory. Long term storage and short term storage.
There is no combat, or pets, or non-housing inventory, or even a three dimensional environment. Just a menu on a different server.
This solution has none of the problems of a bag solution. It is not available during regular play, items are not automatically added, the servers where one interacts with the warehouse are dedicated to that alone.