TruCelt’s friend’s suspiciousness seems to arise not from fundamentalism but hostility to religion so bringing up Nixon would likely backfire.
I see the problem you may have in that the more you insist and argue that it’s not a cult, the more, to a mind primed to see it as a cult, you will be strengthening his confirmation bias.
Have you asked him to read the Wiki page on Quakerism? If he won’t even read the Wiki page, I doubt you’ll convince him.
I don’t suppose he would go to a meeting, either on his own or with you?
Perhaps mention there are non-theist Quakers. That could chip his idea of Quakerism being a cult.
Maybe he confuses Quakers with Shakers? Those are a little strange and lacking in foresight.
Asking him to define a cult might be a good idea. Providing examples of liberal Quakers might help. Drawing parallels with Unitarians (they’re not that far in attitude, are they?) also. Perhaps you could talk about what motivated you to move from Catholicism to Quakerism, that might built common ground.
Or he might have become hypervigilant/paranoid about people around him being stealth bastards and kooks, given some of the political and social developments in the US in the last few years. In which case, you might just have to let him go through whatever process he needs to go through on his own, as unpleasant as that might be.