I was just thinking that with some BPs, they get "stuck" on the prey they prefer, so it can be risky to offer mice if they've been eating rats, & assuming the owner wants to keep them on rats.
If it was a rat snake, or one of many other kinds of snakes, I wouldn't think twice about offering a different kind of hairless rodent, but some BPs will be goofy about it.
Another option: The OP could always shave the rat...you wouldn't have to get all the fur off, just some would likely help.