Yes.
What he will do is complete the 1040 as Married Filing Separately and put down your name and SSN with NRA alongside. I asssume you have a SSN from your time in the USA. He will also list each child and their SSN.
He will then want to use Foreign Tax Credits NOT Foreign Earned Income Exclusion. Doing FTC will allow him to file Form 8812 which is the ADDITIONAL tax credit, which is refundable up to $1,400 per child per year when not resident in the USA. It's actually more if you are resident in the USA.
My recommendation would be to (attempt) to file 2020 taxes using Tax Act, a web based software program. It should bump him into the FTC form automatically and not FEIE. However, the deadline to file taxes as an American abroad was June 15th (December 15th if you filed for an extension), so it may not allow you to file. Worth a shot though. If it works, you guys can work your way through 2020 and then use that as a template for your previous years streamline filing.
If it doesn't work, you'll have to use the paper forms, which are all available on the IRS website. The first year will be the most painful to figure out (will probably take a couple of hours). The other years will be easy once you know which forms you need and how to fill them out. It'll be worth it for $16k though!