The thing is .. the FDIC is already funded by banks.
https://www.fdic.gov/about/what-we-do/index.html
"The FDIC receives no Congressional appropriations - it is funded by premiums that banks and savings associations pay for deposit insurance coverage. The FDIC insures trillions of dollars of deposits in U.S. banks and thrifts - deposits in virtually every bank and savings association in the country."
How the banks choose to pass that cost on to customers/shareholders is up to them.
It's instructive to read https://www.fdic.gov/analysis/quarterly-banking-profile/qbp/... for the last available quarter; the insurance fund is on page 24. I can quote you two numbers and you can decide whether they are big or small: there is $128 billion in the fund, and this covers 1.27% of total US banking deposits.