Kind of true. This is what the bill says:

"No Federal funds may be used for the purpose of implementing, administering, or enforcing any rule…requiring an employee or contractor of any Federal agency or Department to use—(1) another person's preferred pronouns if they are incompatible with such person's sex; or (2) a name other than a person's legal name when referring to such person."

I interpret this to be an attempt to remove any training materials, HR efforts, or resolution conflict mediators from mentioning anything about gender pronouns. The creation of any training documents or resolving any interpersonal conflict where a person calls someone by a name other than their preferred name would be strictly off limits since that time would be paid for by the government. It doesn't matter if there's a concrete rule requiring others to respect preferred pronouns or not. If someone is upset about another person disrespecting preferred pronouns, this bill is broadly worded enough that nothing can be done about it. It creates a loophole for a bad actor to harrass someone with impunity.

This "protects" a harrasser's right to free speech in the same way it protects Nazis from gathering to protest and promote racism. Protecting free speech sounds good on paper until you delve deeper into the implications. Namely, removing limitations on hate speech in order to further suppress others. This bill is not hypocritical, as at first thought, but it is still just what you'd expect from the GOP.