Is there an easy way to trigger automatic failover of a single mirrored database?
If your mirroring is configured correctly for high safety/automatic failover, then I would be inclined to think that you could just take the principal database offline with this:

alter database YourDb set offline
with rollback immediate
go


I believe by taking the database offline, you will have automatic failover without affecting the other two databases that you don't want to failover.