I am a newbie.

I feed my BP pre-killed in a seperate bin from it's main keep.No problems thus far and my snake is easily handled without aggression.I'd like to keep it that way by not intoducing meals in it's main cage.My problem is that it's the first winter I have a snake and while the main cage is set-up to maintain it's habitat even through Buffalo NY's winter,I'm not sure what I should do when the ambient air in my/the snakes room is only 62 degrees F. Now why so cold?- I have cheap landlords and I absolutely cannot raise the thermostat above a certain point. I'm wondering if it's even worth feeding in a seperate bin and expose the snake to those temperatures,even though only briefly,or do I risk illness and excessive stress? I had an idea of heating the feeding bin,but the snake would still be exposed to cold air for a bit to feed and then after feeding. Anyone else going through something similar? Any input on the matter would be appreciated.