Found a really neat bug! One that I am almost hesitant to fix even though I know I should, just because it's cool.
Step 1: Ask Rinny to follow you, thus getting the Companion Rinny status.
Step 2: Do the hypnotism scene with Alissa. Afterwards, you will be deposited back at Cassie's safehouse.
Unlike all the other times you leave Transylvania... Rinny will still be your companion! She'll even participate in combat occasionally! ...but wait there's more!
Step 3: Invite Cassie to the basement, thus getting the Companion Cassie status.
(possibly unnecessary Step 4:) Go to the bed, and it will still have Rinny as the acknowledged companion for the session. Release Rinny and finish the session.
You will go right back to Cassie's spot at the table, with her still there... but she'll also still be your companion! Along with Rinny, still! Unlike Rinny though, far as I can tell Companion Cassie status has no extra effects.
I have also tried to get Linda and Fawn the same way... but Fawn takes precedence over Rinny in the dungeon, and leaves as normal afterward. Linda's behavior in the starting area is also unchanged... though in combat, both she and Rinny will occasionally assist you (though never on the same turn, as far as I can tell).
Haven't yet done anything that would make Rinny leave like normal, but I am guessing that if I did I would then be able to get Cassie to leave as well, and everything would go back to normal.