"should issuing company reissue to avoid a mismatch"

The point is, S Corps do not typically receive a 1099-NEC at all. There are a few professions that might, but the purpose of the W9 is to inform everyone what applies. Your client not only needs to use that form, they need to understand what "operating as an S Corp" means, include how to properly sign papers, payroll, etc. You might have a lot to cover with them.