Good news (I think)
So I took the cat to the vet today and while there I showed them the pics of the goat, and they were not convinced it was anything serious, but left open the possibility. They advised me to tackler her however I could, and try to milk each side. So ... tonight, just now, I got her corralled in the barn, just her and her babies, and ... they were nursing, and she wasn't kicking nearly as much, they kids were latching on and staying on for like ten seconds at a time, and she was tolerating the head-butting milk-stimulating thing they do. Well that seemed like a good sign. So I let that happen for about 10 minutes, and was ready to leave well enough alone, but they told me to check and besides I wanted to make sure.
The fight was not nearly as bad as I feared. I approached slowly, giving her space, letting her maneuver, and then grabbed both horns at the base, as I've read to do. Then it was easier to pull the goat in close (but not biting close), and just hold still and firm, lots of shhhhhh's and calm voice. She struggled, and ran with me for a few seconds, but eventually calmed down and accepted that she was caught. Then I backed slowly her into a corner, and got her left horn with left hand and just stayed still, and just petted her. And miracle of miracles ... she didn't fight one bit when I went for the utter & teats. She was scared, breathing fast eyes wide and unblinking.
The milk was pure white and clean from both teats ... no stink, no pus, no blood. That's a relief.
Then -- and I swear I'm not making this up -- I let go of her horns, and she stayed stock still, and actually let me milk her again. I could pet and rub her (not the ears and face, no sir, don't push it, mister) and I watched as her eyes started to do the more relaxed and slower blinking, which I'm sure means she was calming down.
It was both relieving to know the milk was both flowing and not bad, and that she was letting the kids nurse, and rewarding knowing I went a step further getting her to trust me.