A GFCI extension cord works fine on a non-GFCI outlet.

Yes. That’s literally what they’re for.

A GFCI extension cord has the protection built into the plug end. Plug it into any regular outlet, and everything downstream is protected against ground faults. That’s the whole point—you get GFCI protection where the outlet doesn’t have it.

Just make sure you’re buying a real GFCI cord (look for the test/reset buttons), not just a surge protector. And test the GFCI monthly by pressing the test button—if it doesn’t trip, replace the cord. Otherwise, you’re getting the safety you paid for.