>One side says not using someone's preferred pronouns is intolerant to their gender identity. The other side says being forced to use them is intolerant to their religious beliefs. Now what? Tie goes to the one with the most guns?

Except the paradox of tolerance clearly states that simple disagreement isn't intolerance, nor does it prefer censorship in all cases of disagreement, so your scenario isn't a refutation of it:

    In this formulation, I do not imply, for instance, that we should always suppress
    the utterance of intolerant philosophies; as long as we can counter them by rational
    argument and keep them in check by public opinion, suppression would certainly 
    be most unwise.

What the paradox of tolerance considers to be intolerance, and thus open for censorship, are views which do not allow for rational debate, or respect the existence of opposing viewpoints, but which resort to violent suppression of those viewpoints and the people who hold them:

    But we should claim the right to suppress them if necessary even by force; for it 
    may easily turn out that they are not prepared to meet us on the level of rational 
    argument, but begin by denouncing all argument; they may forbid their followers to
    listen to rational argument, because it is deceptive, and teach them to answer 
    arguments by the use of their fists or pistols. 

That seems like a clear principled line to me.