I think with the "walk the plank" style bucket trap he'll need a pivot point further into the bucket so the rat has to get partway across before it'll fall, which prevents it from jumping back before it falls in. Can do this with a thin rod that goes across attached to the bottom of whatever is used for the plank part.

One method that also sometimes works is putting a ramp on the side of the bucket to let the rat up, smearing peanut butter a few inches below the rim on the inside, and then greasing the edges a bit, so the rat has to try and reach in to get the peanut butter and then slips into the bucket because they can't keep a grip on the sides.

Or he can go the Mouse Hunt route and just befriend the rat. I'm sure that couldn't possibly go wrong.