As you allude to, I don't know if it's clear that the proposed scenario is a clear-cut example. In the United States, people protest against the existence of LGBT+ people all the time, and they are protected by the state to do so.
As others have referenced, it might be helpful to split this question into parts, where first, there is the question of whether hate speech has value in a liberal society, and second, there is the question of whether a governmental body should have authority over regulating hate speech.
For the former, I don't know if it's worth expanding on the answer here since I imagine you're asking more about the latter. But for context of my answer to the latter, I'll name that I personally think hate speech in everyday expression should not be uplifted in a liberal democracy.
However, I am more conflicted on the mechanism through which that should be enforced. The line between social norms and social policy is thin; there are plenty of things we generally understand are "bad" but aren't heavily regulated via policy (e.g., cheating on your girlfriend). Further, I think the definition of speech that incites violence or denies the rights of others is also thin. We can clearly name that protesting against LGBT+ people is intended to deny the rights of LGBT+ people. But what about issues like abortion, or the teaching of black history, or the myriad of other policy issues that affect a given population more than others? I think acting as if there's a simple answer potentially obfuscates the more critical question, which is how we as a society come to understand the values we choose to enforce or not.
Therefore, I think the most direct answer is going to depend on the democratic theory from which you're operating. I myself operate more from the deliberative democracy standpoint, where I am less concerned about the eventual rule than I am with the process through which we define that rule. Using a more liberal version of that theoretical perspective (more on the Guttman end of the spectrum, rather than Habermas), I'd argue the best we can do is make room for speech of all types, but simultaneously do our best to inculcate into citizens the self-determination and liberal values (which, in part, is done by infusing those values in deliberative spaces themselves) that would lead to social norms and systems of power that would resist hate speech much in the same way that explicit regulation would (and could themselves include regulation, as long as it's borne out of a liberal democracy).
Which is my very long way of saying "I don't think there's a definitive answer to this question, but if we rotate the question to focus on how we would as a society identify what speech we will allow or not, and how, we may come to a more robust solution."