Ideas
I have a sim. Every time I try to move her from one lot to another (Using the phone > Move Household) I select "Keep Furniture" on the move. Sim has plenty of money left over. I move, open the lot and all of her possessions that were not already in the inventory are gone.
The items in the household inventory are still there.
I did not have mods on when it happened (I haven't turned them on since 12/2 update), but there were mods in the save. I did have several storage items, like the sticker box and a household keepsake box.
I merged your post with the bug report but to be sure, the furniture from the house is not moved to the household inventory, right ? Only the things that you already had in there when you moved are in there ?
With Keep furniture you should have all the items in the house moved to the household inventory as CGrant said but with this current bug, it seems that the furniture is removed from the house but doesn't appear in the household inventory.
If that's the case, please upvote this bug. If you do have everything in the household inventory that was in your house, please tag me to let me know.