Firstly I wouldn't put a beanie on a baby, it's not safe. You are not meant to be covering their heads once they are indoors.
We have the same issue as you in that we have no temp caps on our heating so it means most of the night it's off so it can be cool. My 6-week-old seems to have adapted quite quickly and doesn't cry in the middle of the night nappy changes. My advice is this:
Make one nappy change at the last night feed (the 10-11pm feed).
Don't change nappy in the middle of the night unless it's a dirty one (quite difficult in the first few weeks but my baby now poos once or twice a day and usually in the day).
If you want to change nappy in the middle of the night, do it quick. Whip off old nappy, give bum a quick wipe and put new nappy on. I used to be meticulous with creaming the baby with my first but a lot more laxed with DS2 and guess what, no nappy rash so far!
At the moment, DS2 sleeps in our bedroom and it's vest, babygro, a fleece swaddling blanket and a merino wrap on really cold nights like this. He's never cold when I unwrap him.
With my PFB, I used one of those cheap fan radiators (only a tenner or get them off freecycle) during nappy changes so at least it's blowing hot air towards him.
Good luck! DS1 was born in Feb and I can say coping with a newborn in December is worse! The winter stretches on...