You have the option to take the exclusion OR the credit ... Choose the credit.

How/where do I report the earnings as there was no company supply w-2. If I select the option not to use FEIE TurboTax takes the info/amount off the misc. W-2. So what’s best way to report this income as foreign earned?
Also, I cannot use foreign tax credit because the country I’m working in doesn’t charge/assess taxes.

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.

If you claim Foreign Tax Credit instead of the Foreign Earned  Income Exclusion, then the Child Tax Credit may still lead to a refund. 

 

However, if you DO NOT take the foreign earned income exclusion this year (with or without the foreign tax credit) you in effect are revoking the foreign earned income exclusion and you must wait 6 year to be able to use it again.

 

According to the IRS: Once you choose to exclude your foreign earned income and/or your foreign housing costs, that choice remains in effect for that year and all later years unless you Revoke it.  See Publication 54, Tax Guide for U.S. Citizens and Resident Aliens Abroad for instructions on how to revoke the election and the associated effects.