You can book by phone at SNCF. No need of Pass Cover Number. 

https://www.sncf.com/en/customer-service/contact-us/telephone

 Press #85 for English, no booking fees, reservations are sent via e-mail.

I asked the question to SNCF and their answer was this: “Interrail pass holders can make seat reservations on the train only via Interrail (on the website or the Rail Planner app) or, if it is not possible to book in this way, at the station. There is no way to reserve your seat on our site using your Interrail pass, it is a service that is entirely up to them and not the carrier.” By booking the ticket through that phone number, how can I pay the seat reservation cost of € 10 instead of the total price of the Nice-Paris route?

You can't make the reservation at the SNCF site but you have to make the reservation over the phone. Call again. Sometimes the person answering is not so helpful. At least a few days ago pepole have made reservation by calling SNCF. 

I called SNCF but was told they only accept American Express, French or German cards. I can't make a reservation if I don't provide the card, how can I solve it? I have a card but it is from an Italian banking institution. How do other countries make a reservation?

That is a lie.

That is also a lie.

Multiple people have used the SNCF phone service over the past few months, from many other countries. You're not the first one though to report such reactions from SNCF.

It sounds like SNCF (or maybe some of their employees) are trying to get rid of people who want to make Interrail reservations. But the main reason people are calling them is because SNCF themselves removed the booking possibility for pass holders from their website.

I understand that these are lies but I called 7 times talking to 7 different operators, each of them reiterated that they only accept that type of cards, otherwise I have to contact Interrail or go to the station, which is impossible for me since France is a destination of the trip, not my country of residence. I'm in a dead end.

I very much doubt it is allowed to accept credit cards from some EU countries and refuse them from others. That is probably a breach of the SEPA regulation. But that doesn't help you now of course.

You can also call Belgian railways. Their number costs more but is otherwise fine: no booking fees, no pass cover number, reservations are sent via e-mail. And they can also help you in Dutch and German :)