I don't believe the Fidelis consider themselves Clan, nor would the Council of 6.
In the ending of Surrender your Dreams Paul Moon gives the next in line a book with a Smoke Jaguar on the front of it and tells them to go claim their birthright by stepping on Terra without fanfare.
The Fidelis Crest is a daggerstar on a shield.
In the beginning of the second succession war book information is being made available to them by the Exarch. This suggests that they might serve the Exarch. They also refer to the time that the Clans are not in the Inner Sphere as their Exile.
It is unlikely that Clan Wolf or Clan Jade Falcon would serve the Exarch. But the Fidelis already do.
The Smoke Jaguars weren't abjured. They were, "Annihilated" it obviously didn't stick. They'd just need to win a trial against anybody who said they weren't a Clan to prove it.
Clan Wolf and Clan Jade Falcon believe they can win and have to defeat them anyways to take Terra. So they'll accept a trial if it's offered.
The Fidelis is hardly Clan Smoke Jaguar anymore. So from a game design perspective it neatly gets around any Clan "Winning" and giving players bragging rights over another group of fans. It also can couple the Republic and the Clans thus reducing the number of factions.
It's an optimal solution.