nobody is saying they can't use the name, the government just isn't obligated to give them legal protection from other people using it.
So in essence one could argue trademarks are government speech and it can refuse to endorse it.
I think the only way someone could come to this is a mix of the 14th and 1st. But courts have already restricted speech in places like public airwaves because the government has a place in endorsing it.
This is all kinds of horrible reasoning. Again they could go the 14th+1st amendment root but just the first amendment root seems counter to so many precedents on the government being able to restrict speech if it becomes government speech.
OK, first,
FCC v. Pacifica Foundation does not say what you think it says. Besides, both it and much of
Red Lion Broadcasting Co. v. FCC were bad law when written and remain bad law now. The reasoning allowing regulation of indecency rather than just obscenity in
Pacifica was "First, the broadcast media have established a uniquely pervasive presence in the lives of all Americans." and "Second, broadcasting is uniquely accessible to children, even those too young to read." It was under this line of reasoning that the Court justified the FCC being able to regulate indecency rather than just obscenity. Justice Brennan's dissent on the case does a wonderful job pointing out the huge flaws of the majority opinion.
As you may have noted above, the constitutional ruling in the case had absolutely nothing to do with any idea of government speech.
Second, the 14th Amendment has nothing to do with this case as the government entities, rights, and statutes involved are all entirely federal and therefore don't require the 14th for any sort of incorporation.
