Yes, getting some milk into the kid... even if it is cow milk that tastes funny to it and comes from a bottle is #1... make sure the babe has enough to eat, don’t starve it thinking the kid and it’s adopive mom will suddenly get back to normal without any help. Priority #2 is figuring out what’s going on with the udder that there was the sudden change. Is it swollen and hot? Any redness on the teats, signs of pus or sores?
Do you know how to milk her?
Separating them but within sight/smell might be a good way to see if she truly is dry, go out first thing the next day and try to milk her. If babe is separated for a night she should be full the next morning if she is still in milk. They don’t usually just go dry overnight, so if babe has been sneakily sucking you will see it.
If that’s the case and there’s no udder problems you may have to halter and hobble tie her and assist in nursing, basically stand over the nursing and force mama to let baby nurse for a while (had a sheep kill one of her own twins and we had to do this with them for two weeks before she accepted her other one... also a cull this year)
If there is a more experienced goat owner you can consult with they might be able to look at her more closely and advise you about what’s going on. More pictures of the udder, maybe the goat and the kid as well would be helpful. If it is mastitis you will need to treat with antibiotics, or risk losing the doe or having her lose the ability to produce milk again.
Observe carefully and often. With bottle feeding, feed frequently in small quantities (the cow milk is harder for them to digest and switching could take some time)! Start with 5 times a day with fairly even spacing and go from there.
Best of luck, and I look forward to an update where this is resolved for you! Patience, persistence, and careful observation are your best friends right now.