When the Direct chat begins, once the two rooms are created, they are plumbed together. Whichever of Alice's accounts come and go through the room, from Bob's end, it's always a 'chat with Bob' at her end, and a 'chat with Alice' at his. With the appropriate name and avatar at each end.
The second solution is potentially fiddlier, because it tinkers with the sessions verification system...
(5/?)