I'm not aware of any obligation to use either the FEIE or the Foreign Tax Credit. Given your situation with 4 kids, using the FEIE automatically disqualifies you from claiming the refundable Additional CTC. You are most likely better off not excluding any foreign income. If you don't exclude your income as foreign income, your tax liability will most likely be eliminated with the CTC and you would receive a check for your eligible ACTC. There is not a requirement to live in the U.S. to claim these credits.
