How much are you planning on clipping off? If she is in the stanchion with grain, and you are doing her rear legs, udder, and rear end area she should be fine I would think. It probably won't take you very long.
I wouldn't clip her whole belly.. maybe just about halfway, and even that could be a bit much. Definitely do her tail and around her vulva area, and her back legs and udder. Those are the most important areas, and they will get the most messy .
But I'm also going to be milking her so that's why I'd be doing part of the belly. Should I wash her backside a little bit, so that the clippers don't get clogged? I could take her inside to do that but I don't want to freak her out.
I shave the tail the backs and insides of the rear legs and the udder to just in front of it. Sometimes my does get long hair we call wings on their knees on the fronts of the back legs and i tril that off usually with the hair not against for that. I have seen babies get confused and suck on long hair rather then teats. and if your milking its nice to have a clean cut.
i just think with the cold weather you are having shaving her belly wouldn't be a good idea. Just in front of the udder like Beth suggested is ok I guess - I just worry about my goats getting sick when they need that fluff
If she has been nursing on her mama for that long, it is probably going to be hard to transfer her to the bottle and it will also probably be stressful for both the baby and mom. If you are going to be bottle raising it, it would be best to do it from the beginning.