Just to throw out another possibility, although the best approach really depends on the qualities of the specific foam you are working with, you might consider sharpening the mouth of a piece of steel piping and using that. With a softer metal like brass you can actually sharpen it by hand with an exacto, but steel you’ll likely need to use a lathe to cut a sharp edge.
I had to cut holes some ~10mm felt recently, and any conventional drill bit would not have worked because it would catch on the material. So I chucked up a steel pipe with a sharpened edge in the drill press and it worked quite well. This is also a good tool for cutting holes in something like foam core. You can also make cuts with it free hand, twisting it back and forth.
Of course if you aren’t going all the way through you’ll need to find a way to cleanly trim out the waste.
Note though you might need some kind of core in the pipe ‘bit’ you fashion, so that it isn’t deformed when you chuck it in the drill press.
The soak and freeze method sounds pretty cool, never heard of that!