It is absolutely ok to leave it in the cage overnight to see if he'll eat it. However, if you leave it out overnight, then I would not recommend refreezing it.

Whether or not you refreeze depends on how well it was frozen and thawed to begin with. If it maintained most of its bodily integrity after being thawed (looked and felt mostly like a freshly killed rat) and it's only been left out for an hour or two, then I don't think there is an issue with refreezing. However, if the belly has swelled up, or if the middle of the rat feels particularly mushy, I would not recommend refreezing.

Each time you freeze it, and the longer it is left out while thawed, the more the tissues break down...which can lead to an exceptionally nasty exploding rodent the next time you thaw. So my rule of thumb is to only refreeze if it still seems to be in really good condition to begin with. When in doubt, throw it out.