That's not possible online as far as I know.

You could contact GWR directly and ask whether they can book a seat for you on their night train.

I contacted GWR by email and got an answer within a few hours. That answer is in line with what was already posted above by Railmaster! Thank you Sir. 

Below I join the GWR answer :

“Dear Mr,

Thank you for getting in touch about the Night Riviera Service.

For customers holding an interrail pass, you will need to contact us by phone on 03457 000125. Seat reservations are free and subject to availability, however if you require a cabin this will need to be purchased as it is a journey add-on. Our team will be happy to assist with booking this.

Thank you for contacting GWR. If you have any further questions, please do not hesitate to let me know. 

Your sincerely

Andrei
Retail Customer Service Advisor
Web Support Team
Great Western Railway

It certainly is possible to book a free seat reservation on the Night Riviera from the GWR website or app

When logged in go to bookings-seat/bike reservation.

Select your train then select “anytime ticket” (this is only to confirm you hold a ticket for the train, not to actually buy the ticket), there is another step or two and it will then give you your reservation.

AFAIK there are no allocated seats on this train, just reservations limited by the available number of seats.

Although it is listed as reservation compulsory you would likely only be refused travel if you turned up without a reservation and the train was fully booked.

Nice to hear that solution. Meanwhile a friend of my advised me to try ACP Rail website what I just did. It was peace of cake to reserve a seat. Only 2 differences, I had to pay 6 Eur incl 2.5 Eur booking fee for Britrail but with my hearing problems, calling is always a complex procedure I prefer to avoid. And calling outside EU would also have a cost depending how fast you get somebody on the line and the time it takes to complete the whole procedure. On the other side, in the PDF reservation file I got by email, there is no car and no seat assigned; is very strange for a reservation.  On its home page ACP says this about it : “Buying a Seat Reservation in the UK does not always assure an assigned seat. Instead, it will guarantee an available seat on the train.” 

How I will find that available seat is a good question. I hope I will not get problems in the mid of the night.  Thanks anyway

To be honest, ACP is best avoided. They indeed charge €6 for free reservations.

To be avoided for the 6 Eur charges or  are their bookings not reliable? I always prefer booking directly on the websites of the railway companies or through the interrail/Eurrail app but when not possible, 6 Eur is not the end of the world. Most railway companies charge for their work, and sometimes a lot more than 6 Eur. 

Because of the charges. I haven't heard of any problems with their reservations, other than people being upset after finding out that they've paid for something that is available free of charge

The rail planner app can't book anything, it just forwards to some of the websites where you can book, including the Interrail/Eurail websites. I also avoid those if possible, mainly because of the extra charges. 1 reservation with an extra charge is not the end of the world, but if you're doing multiple reservations with multiple people, then it adds up quickly.

Thanks for explaining, Railmaster!