You are garbling two different provision. One is the "Foreign Tax Credit". That is a credit that is based on taxes that you paid to a foreign country. If your client is not paying taxes to a foreign country, that does not apply.
The second provision is the "Foreign Earned Income Exclusion". That is for a person whose "Tax Home" is in a foreign country, and if that person is outside of the US for at least 330 days in a 12 month period (or meets the Bona Fide Residence test). There is no requirement to pay taxes to a foreign country. So your client may qualify for that.
Just be aware that if your client qualifies to "exclude" the foreign earned income (Form 2555), that doesn't mean it doesn't affect their US tax return. If they have other income on the tax return, using the Foreign Earned Income Exclusion essentially pushes the taxable income in a higher tax bracket.
