The cash advance problem is a Visa problem, so you'll probably see the same issue with the CSP. Luckily Plastiq warns you that the transaction will process as a cash advance before charging your card.
You can do as
@Angrose suggested and set the cash advance limit on your card(s) to $0, or as close to zero as the card will allow and make a small payment that processes as a cash advance to establish the relationship with the payee, then charge the balance as a credit card transaction.
Or you can establish a relationship with the payee by first paying with an Amex, which will not process as a cash transaction, and then make subsequent payments with a Visa without problems.
ETA: Of course, you'll want to use a Visa to pay mortgage or rent via Plastiq because it'll code as 2x travel on a CSP and 3x travel on a CSR/CIP (at least for now). Paying with an Amex will not trigger any travel bonus category.