Which banks have $200 ATM limits? Citibank? JP Morgan Chase? Bank of America? Wells Fargo? US Bancorp? PNC Financial Services?
There are always going to be small banks that suffer in a downturn but they aren't going to have a significant impact.
The Fed will keep.increasing interest rates until the job market weakens, most analysts are now forecasting a 6% base rate. The difficult tasks is not crashing the job market, if they manage that then there won't be any meltdown as you suggested.