Do you have any extra kids coats (for the pybmies) and one of your coats hanging around that you could possible use on the goats for a day? When I bred my doe last month I forgot her blanket and I let her wear my coat on the way home. Washed the buck smell off and it is back in use as my farm coat. I transported her in the back of a pickup in a crate (unfortunately without a tarp). It was about 30 out that day. She made the trip just fine after I folded my sleaves up.
ETA: picture
Mom and Ethanol
ETA: picture
Mom and Ethanol