If a Jack Russell is unavailable, tie bread onto the trap with thread. It sounds horrifying, but it gets caught in their teeth & ... well... desired outcome becomes reality.

If you have a dog you'd like to train as a ratter, you can get humane traps from bunnings (square 'pipes' that are bent in the middle, and have a trap door), and after trapping the rat/mouse, take the dog & the mouse-in-trap into bathroom. It's somewhat of a blood sport & not for those with any major feelings towards rodents, but it trained our dogs very well.