The availability of passholder seats on the Eurostar can be checked here:

https://www.b-europe.com/EN/Booking/Pass#Outbound

On the 10th of July there are only passholder seats left on the 16.43 train, see picture below. 

If you don't have your Pass Cover Number you should be able to book by phone at SNCF. No need of Pass Cover Number. 

https://www.sncf.com/en/customer-service/contact-us/telephone

 Press #85 for English, no booking fees, reservations are sent via e-mail.

@Tubby 

The international phone number is in the link I posted. Just look a bit down.