You know what I do for a living? I work in risk management with major banks. Believe me when I tell you banks are very much subject to all kinds of state laws. They can be inspected by state auditors. Cuomo when he was governor of NY threatened the banks with exactly that in order to get them to support gun grabbing. If NY can do it, other states can lean on them in the opposite direction using the exact same means.
Are the banks you work for chartered by the state banking department or the OCC?
The state banking departments or credit union regulators have zero authority over Federally chartered institutions. The Texas Department of Banking can enter a national bank but have no more authority than that agency would have over a sheet metal fabricator or a wholesale food distributor. Only the OCC and the Federal Reserve (in the latter case if the national bank has a bank holding company) have the authority to review the records of a national bank or a Federally chartered savings association.
A national bank or other Federally chartered financial institution is subject to state taxing authority or the state corporation commission if the Federally chartered financial institution has a bank holding company. If you are at a national bank, you may be seeing auditors from the state tax authority.
Nevertheless, state governments do have leverage over national banks because state agencies and pension funds have enormous deposit funds on which they can set conditions. There is also the publicity leverage; in a liberal state like New York, there is a lot of support for gun grabbing. Conversely, conservative states like Texas and Florida have used their influence to discourage banks from supporting ESG initiatives.