Start with the idea that there is no such thing as a "BSA Hold."
Someone in your bank is a bearcat for nerve if they believe they have any right to hold the proceeds of an incoming wire without significant evidence of wrongdoing.
Regardless of who reaches out for additional information, it makes very little sense to contact the originating bank. Cautious originating banks sometimes require copies of invoices, but I've never heard of a beneficiary bank making such a request. The originating bank probably will not have the information and might reasonably choose not to make it available to you if they did.
It's your customer whom you should be talking to.
I can make an argument that the people who are going to be evaluating the answers should be the ones asking the questions so we don't have a lot of "Well, go back now and ask them this." However, that's a common problem and I cannot resolve it.