The pass holder seats sell out on the Eurostar before the train is full. That's why you can see seats on the Eurostar website but not on eurail.com.

There is a limited number of pass holder seats on the Eurostar between London and mainland Europe. On popular departures and during high season those sell out weeks, and sometimes months in advance.

The best place to see the availability of passholder seats and make reservations on the Eurostar is 

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

If you don't get any result, press "later trains" and eventually you will find the next available connection. 

If you have a mobile pass you need to generate a Pass Cover Number in order to make the reservation at b-europe. You do that here in the PCN generator:

https://community.eurail.com/news-and-announcements-39/pass-cover-number-generator-is-live-5653

What route do you want to travel? Do you have 1st or 2nd class passes?

There is a quota for passholders on Eurostar trains. It can sell out weeks in advance. There are no available trains to Paris that day for passholders and most trains are fully sold out anyway.

Here is the only possible option on 26th June for 6 people (with 2nd class pass) :

- Eurostar London St. Pancras - Lille-Europe 17:04 - 19:26 30€

- easy 5-10 min walk

- TGV Lille-Flandres - Paris-Nord 20:12 - 21:14 10€

Book Eurostar ASAP through Anna's link to save booking fees. Book TGV here : https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish

Afterwards your trip should be much easier. Still book TGVs quite some time in advance, they do sell out (no quota). Do not hesitate for questions.

The only other option would be to spend a night in Lille (either on Sunday or Monday) so not really what you want I think.