I tried not to respond because I don't want to contradict too badly, AND confuse matters worse, but if the barn is good and tight, not drafty, the doe is healthy and she and any other animals out there are not having any problems now in the barn, it may not be best to bring her into a place like a house where the chaos of the move would stress her more.
Is it above freezing out there? Water not freezing up? Are any of the animals shivering, looking uncomfortable? Any sign that they're in any distress from the cold?
I'd be watching her VERY close, especially since you say there's no way of bringing a heat lamp or anything like it in.... making sure the kids get dried off quickly when they're born etc.... polar fleece is very washable and very warm even if it gets wet, unlike cotton fabric which looses any insulation properties if it gets wet.
If you need to bring them in after, so be it, but stressing mom out now is probably not the best thing. Of course, you know your situation best, if she's not easily bothered by getting lead into the family room, well, that might be just fine.
I'm sorry if this makes it harder... I don't mean to.