That's what I'd do- put the rat in some well-used (dirty-smelly) mouse bedding first, to transfer the scent. A f/t mouse won't have much scent- use the bedding!
It's a bigger jump going from live mouse to f/t rat also- your best bet would be a live baby rat (small & eyes closed!) first, with the mouse scent transferred right before offering the meal.
Switch to f/t rats only after one or more live baby rats are easily taken (use your judgement on that), as it's just hard to convince some snakes using tongs to jiggle the prey just right- & you need the snake to accept the different flavor. Rats don't smell or taste* the same as mice- not at all. (*so they tell me)
If you try just a f/t rat first, with mouse scent, you also need to warm it as suggested- like w/ hairdryer, to lifelike warmth, especially on the head.