So I finally found the rules about creatures that take up more than one square and falling, it was in “Forced Movement and Terrain” of all places. Turns out, if you can’t push every square the creature occupies off a ledge, then you can’t push the creature off the ledge. Even if it’s a 5×5 monster, if it’s got one toehold on a corner square, it doesn’t fall. It is forced to either squeeze or move back on its next turn, but it does not need to save, does not fall, and it is not prone on the edge.
Interestingly, on the same page, it says forced movement ignore difficult terrain, but not challenging terrain. So if the mud ice is just slick enough to take extra moves, pushes ignore the effect, but if the ice is slick enough to force checks to avoid falling down, then if someone pushes you onto the ice, you have to make a check to avoid falling down. Totally makes sense, really, and makes forced movement powers a lot more fun (if the DM ever uses challenging terrain, which I haven’t seen often. We tend to forget the category exists.)