If you nicely ask the FA after a while for another drink, you might get it. Especially if you're in the back for a bathroom break already and ask there. Of course some FAs may mother a kid that flies by themselves a little and cut them off from the sugar.
BTW, I inquired about UA only accepting credit cards for snacks and alcohol. I inquired what would happen if a child did not happen to have a credit card for a snack purchase. The FA later informed me that he found out their card machine has a button for kids (cash purchases?). So I guess the credit/debit card policy isn't totally set in stone.