This reservation can only be made online through the Interrail/Eurail website, see the reply in the link.

https://community.eurail.com/eurail-interrail-passes-41/log-in-12872?postid=59118#post59118

Updated: Seems that 2nd class of the train is fully booked (at least from Strasbourg). The train couples to another set in Strasbourg, so from there capacity is increased and still places to book on the other train set (which leaves at the same time, same train).

You can always take reservation free regional trains from Karlsruhe to Strasbourg, and take the train from there by booking from.

The same TGV leaves at 8:20 in Strasbourg.

• Leave Karlsruhe at 06:06 to Appenweier (RE2 to Konstanz). Change in Appenweier, arrival 06:47
• Leave Appenweier at 07:40 (RB25 to Strasbourg), arrival Strasbourg 08:04.
• Take the TGV at 8:20 in Strasbourg to arrive in Paris at 10:05 (reserve this stretch of the train via https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish for which you need a pass cover number ).

Later on the day there are better connections though, with more convenient connections and less time in smaller stations in early morning. (Usually change in Offernburg) If you have the time for that, book a later train in Strasbourg, use regionals,  and sleep a bit longer.
 

Trains about every 2 hours in Strasbourg to Paris, hourly at mornings and evenings.

Looks like I'm able to book the direct TGV now for 17€+2€ through interrail.eu. Should be the same on eurail.com

In any case book ASAP.

Let us know !

Edit : per sncf-connect.com (only to check availability) there are 5 seats left, maybe a family cancelled its booking :)

Thank you so much!!! I got it. And thanks Brendan for your insider info! First time in Paris thanks to you GOATs.