To me, this is another situation where game mechanics are causing issues to story and I'd rule based upon what makes most sense from the story's perspective.
The story is this character has an ability to potentially cause a beguiling twist to a target. They know they can accomplish this thing... this beguiling twist. It is one of their abilities, right there on their sheet. It is something they can do in the fiction.
However, the way the game mechanics have been designed, there's in theory no way they can use this ability-- immunity to frightened condition means no saving throw... no saving throw means no way to trigger the reaction for beguiling twist. Game mechanics are prohibiting the use of an ability they should be able to use for no other reason than just the way the game rules are getting combined together. That's a fault in the game rules, not a fault in the character. And thus I won't punish the character because the game rules are causing a problem. I'll adjust the game rules.
So for me... this is where story trumps mechanics and I do what I think makes most sense story-first. To me, the character has a chance to cause a beguiling twist as a reaction when they are exposed to something that could cause fright. To achieve this using the mechanics... I would let them make the saving throw roll just to discover the number that pops up, and whether that number is high enough to cause the reaction of the beguiling twist. But regardless of what number rolled... the character will never be frightened because they are immune to the frightened condition. Doesn't matter what the saving throw was, they are immune to frightened. But the roll at least gives us an indication of when the character can use the ability as listed.
That's how I'd rule it.