Should work just fine. I wouldn't worry about the foam, if you drop it in a 4" hole its going to go down under the ice before it pops back up.. the odds of it coming straight back up a 4.5" hole isn't great enough for me to use an uncomfortable scooper with a bunch of foam on it.
Or you could just push the auger down in the hole and then pull it straight up hard and fast (with BOTH hands on the auger or the handle will nutt check ya). Can't remember the last time I have used a scooper after figuring out that trick. With the power auger I just pull it up with the engine rev'd up and it clears all the chips.