The pass holder seat on the Eurostar sell out weeks in advance. The first available pass holder seats for 4 persons, in 2nd class, from London to Paris are on the 6th of July.
The best way to make reservations for French trains is to call 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
To get your Pass Cover Number, if you still will need it, you need to contact Customer Support through this form and ask for the Pass Cover Number.
https://eurail.zendesk.com/hc/en-001/requests/new
Customer Support is currently overloaded with requests so you will probably have to be patient to get help. Please let Customer Support know what date your travel will start so that they can prioritise your request correctly.