The FDIC’s balance sheet is unlimited. It’s an explicit full faith and credit agency. It has, to date, never pulled on that credit by managing its risk through fees on member banks.
If so, interesting, I always sort of assumed the money would just come from the treasury at that point.
Last year's failures in March caused an assessment in December [1]. In a crisis, that isn't enough time.
The way the system is designed, the Treasury would write a cheque. Full faith and credit means those FDIC claims would have the same legal standing as e.g. Social Security payments or tax refunds. Treasury would probably structure it as a loan to be paid back by future assessments, again, for political reasons. But they'd have discretion on the specifics at that point.
[1] https://www.fdic.gov/deposit/insurance/assessments/specialas...