You still should have put the kits in with momma that morning. As long as they have bedding & fur they will be fine in the cold weather because they should be covered up and snuggling together.
Denying them feeding time with mom because of the weather isn't "overprotective", but rather your starving them and once again do not have their best interest at heart.
You could have put them out with mom, or at least brought her it...at any rate they should be with her at all times. She isn't going to harm them. She is taking great care of them, but by you only giving her the nestbox at certain times, you are not allowing her to feed them on a scheduled time, and you might make them miss their feeding time.
Please put the nestbox in with momma and leave it in there with her.
What does your setup for the doe look like? Does she have a hutch that is covered - because that should keep the rain out.
Please put your location in your profile. That will give us a better understanding of what type of weather you might be dealing with.
I honestly hope you do not breed her again. You say you have the kits best interests at heart, but you don't. You should be letting them be with momma all the time. Not just letting her see them for a few hours every day...that's not right.
Emily