Who are "we", in relation to "them" (the bank)? .... Are you an external advisor, ....Internal Audit?
I think that they should document every correspondent bank they have an account with - using either something such as audited financials that demonstrate that each bank is a US chartered bank, or failing that with documentation from a reliable, published, source, e.g. an entry in the Banker's Almanac.
These records would/ should then be sampled (or reviewed in entirety if there are relatively few correspondent banks) by Internal Audit and visiting regulators.
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This is my opinion; it is not legal advice, nor the view of my employer, and it may change tomorrow.