Today it isn't possible to choose between sites in multiple resource databases on one server, in Verse on Premises. You only get the site(s) of the first (?) r&r database proposed, while looking for an available resource / room.
Multiple R&R databases are supported in Notes, iNotes, but that doesn't seem the case in Verse.
In our case, this is a move-to-Verse stopper and should be fixed (alternatively we could live with a procedure to migrate the rooms, resources & sites to 1 database, but I don't think there's a procedure for that).
To be clear: the issue exists with multiple resource databases, it doesn't exist when you have multiple sites in just one database. The following KB isn't very clear about that...