Your kids are precious!
We've spent a couple of Christmas/New Years in Disney with the most recent being last year. On both trips the weather ranged from perfect (sunny and warm, but not HOT) to very cold (layers, hats, gloves, and still shivering). Last year there were days that the weather dropped into the 30's overnight and didn't warm up much during the day. My kids swam on Christmas Day, though, so you've got to be prepared for anything.
We spent both of the NYEs we were in the World at the MK, but I have read and remember hearing last year that they were checking IDs at the Poly Beach and not allowing non-Poly guests access unless they had ADRs for the evening. Just thought I would throw that out there so you can verify before dealing with transportation and then realizing you can't do what you've planned to do.
I love your laid back plan!