9001:2015 8.4 Control of externally provided processes, products and services

It seems to me you have two problems. The first is that your supplier approval process is too rigid, and the second if finding a supplier which will meet your customers requirements.

The first is relatively straighforward. We had a similar problem and wrote in some clauses which gave us a bit of leeway, eg, a Director could sign a concession for a particular supplier / individual PO if there was really no other option but to use them. We generally do insist on proof of insurances in any case.

The second problem is more knotty. Can we assume that you have reviewed your design to see if a different material or technology could be used even if design needed to change to facilitate that. I would also be contacting the customer to discuss the problem and ask if they would be prepared to accept a concession on the documentation, or if they know of a supplier who could meet their requirements.