We don't normally put our does in the kidding stalls until after they kid, I like them to stay with the herd and feel as natural as possible. So ours are really post kidding stalls, but on occasion I have made a double size pen and put one in that was having problems or seeming distracted by everything going on.
In the winter if the kids need the heatlamps I leave them in 2 or 3 days. If the doe is not a good mom, or baby had weak legs longer, until baby is up moving around really well. Sometimes I never put them in the kidding pen, if the weather is nicer and all is going well.
Unless you have good sized full-size breed, I think the babies would come out of the 4x4 panel, I had a 2 week old kid the other day, squirm through the panel when we were moving animals around, and we have full-size goats.
I know greendecember on here put chicken wire on her fencing near her barn, because some of her kids were going through her panels.
You could try the 4x4 and see how it goes, never hurts to try. But might be prepared to have a back up plan.