Hello Paolo, Yes, you could reserve seats without activating your pass. Please also note you don’t need to book trains in advance in many countries, unless you want to reserve a seat, e.g. Germany. If you have EUrail pass, you can directly reserve seat with this link: https://www.eurail.com/en/book-reservations#/For the Interrail pass holders, I think you can also easily find the corresponding link with Google.
The best place to check the availability of pass holder seats on the Eurostar is here: https://www.b-europe.com/EN/Booking/Pass#TravelWish The first available pass holder seat from London to Paris is on the 13th of July, see below. Sorry I may post a wrong website. Anyways, in the link you gave, there are still some tickets available on 6 July
If it is not a very urgent issue, I think you could submit a request here to fix your issue https://eurail.zendesk.com/hc/en-001/requests/new
Maybe you could call SNCF to book your tickets https://www.sncf.com/en/customer-service/contact-us/telephone
Could you please check whether you can book tickets via this website: https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish ?I just checked that there are still some tickets available on the day you chose.
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