She may very well drink tonight, so I might wait until morning, but probably not any longer. If she still has it in her mouth by then I'll try and give her a soak to get it out. I feel like that will risk less of her vomiting up the rat and will probably not be too risky for an infection?

Yeah, I need to figure out a good solution for her because she tends to press the rat into the substrate in order to get it all the way into her mouth. As far as I know that's where the problem is. She tends to go kind of far and wide when she does this, though, so probably would be newspaper and try to weigh it down somehow?