Just played a game with Dungeon set aside by Way of the Mouse. When playing Bridge Troll using Way of the Mouse, the Bridge Troll would discard itself at the end of the turn. I would still receive the Dungeon benefit at the start of the next turn, but the Bridge Troll is already in my discard (and occasionally back in my hand), and I did not receive the cost deduction benefit. This seems like a bug to me?
Yes, that's a (known) bug. Generally, when a Duration card is played using Way of the Mouse playing another duration, it will not stay in play, though it usually should.