I would imagine this ordinance probably would be upheld as a reasonable time, place, or manner restriction of free speech.
Well, I'm not certain that it would even be considered speech under the 1st Amendment. The seminal flag burning case stated that "[t]he First Amendment literally forbids the abridgment only of 'speech,' but we have long recognized that its protection does not end at the spoken or written word. While we have rejected 'the view that an apparently limitless variety of conduct can be labeled `speech' whenever the person engaging in the conduct intends thereby to express an idea,' United States v. O'Brien, supra, at 376, we have acknowledged that conduct may be 'sufficiently imbued with elements of communication to fall within the scope of the First and Fourteenth Amendments,'" Texas v. Johnson, 491 U.S. 397, 404 (1989) Based upon the factors listed thereafter in the case I'm not sure the conduct would reach the level required for speech. If however, it were determined speech the question would then be "whether the State's regulation is related to the suppression of free expression." Texas v. Johnson, 491 U.S. 397, 403 (1989); see also United States v. O'Brien, 391 U.S. 367, 377 (1968).
The key issues would be that it is likely not even considered speech and if it is speech is not prohibited because of the State is attempting to suppress speech but rather the ordinance is for safety purposes. Similar to the fact that while burning a flag may be protected free speech and a law prohibiting burning flags would be a violation of the 1st Amendment a law that prohibits burning flags in a public building (while having no prohibition against burning flags in other places) would not be a violation of the 1st Amendment. Partially because as you stated time, place, manner restrictions but also largely because the law is not to suppress speech but rather for public safety concerns.
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