Last information is that Frecciarossa trains in France do not accept Interrail and Eurail in the moment. You can use the TGVs on the same route, but the reservation is about 30€. 

Very interesting, that italiarail allow to book this. 

So I shouldn’t try to book it by Italiarail even though it allows me to?
Unfortunately it’s just not an option for me to travel with TGV.

It doesn't really allow it. As soon as you click/tap on "Add to itinerary”, you'll see an error message:

So you can't book it.

Italiarail does allow to book this train for domestic journeys in Italy and interestingly, their booking fee is only 1 € per order (I also tested 2 passengers):

Very interesting, that italiarail allow to book this. 

It doesn't really allow it. As soon as you click/tap on "Add to itinerary”, you'll see an error message:

So you can't book it.

I can do it using this exact link:
https://partners.italiarail.com/default?force_pass=true&aff=EURAIL
The timetables are valid until June 11 if I remeber correctly

It doesn't really allow it. As soon as you click/tap on "Add to itinerary”, you'll see an error message:

So you can't book it.

I can do it using this exact link:
https://partners.italiarail.com/default?force_pass=true&aff=EURAIL
The timetables are valid until June 11 if I remeber correctly

That's the one I used :)

Don't you see this message when you click on "Add to itinerary”? If not, on which date did you try? I tried on 6 April.

The TGV should be around the same time (this is on 6 April):

But as you can see, reservation is quite expensive (33€, including 2€ booking fee).

It doesn't really allow it. As soon as you click/tap on "Add to itinerary”, you'll see an error message:

So you can't book it.

I can do it using this exact link:
https://partners.italiarail.com/default?force_pass=true&aff=EURAIL
The timetables are valid until June 11 if I remeber correctly

That's the one I used :)

Don't you see this message when you click on "Add to itinerary”? If not, on which date did you try? I tried on 6 April.

I tried on 1 Jun 2022.

So I shouldn’t try to book it by Italiarail even though it allows me to?
Unfortunately it’s just not an option for me to travel with TGV.

The TGV should be around the same time (this is on 6 April):

But as you can see, reservation is quite expensive (33€, including 2€ booking fee).

Thanks! I will have a look at that one. 
I also received a message from Italiarail which says:
“Hello,
Italiarail currently has no access to passholder fares, as we are in the process of upgrading 
to a new booking engine with the train line and they have not granted access.
You may wish to try one of the other vendors like raileurope.com or omio.com to see if you can book”

I had trouble reserving Italian tickets yesterday (for Milano → Rome), maybe a technical glitch, but in the end https://tickets.oebb.at/ worked best.

ÖBB can’t sell online reservation for italian highspeed trains. Maybe you take for the nightrain to Vienna/Munich? 

ÖBB can sell online full price tickets of Trenitalia trains.