Fast and done would be my vote.
Goats are livestock animals, not pets imho. Livestock are supposed to generate income in some sense, not drain it endlessly.
I would tell my kids that - of course - the baby needs to stay with its mommy, and that their little family is going to be better served at a new property with people who have the time and resources to deal with them.
On a side note, goats are a major pain in the butt. My job as a kid was keeping our goats out of my dad's garden. They would eat anything and could squeeze through the strangest places. Ours stunk to high heaven too.
Never been a goat fan so take my response with a grain of salt.
Good luck.