Given that that new polar region would get way less sun than either of the existing poles, the icebergs below it might grow a lot higher than they do at the (ant)arctic.

Of course, there are way too many unknowns here. Icebergs will collapse under their own weight at some height, but gravity from the moon would somewhat counteract that. I also wonder what weather patterns would look like, whether stalactites could form on the moon, etc.

Maybe, we could get ice to connect earth and moon before the moon fell down to earth.