Erin, you seem to be dodging the difficult questions here. If you really want to convince people to think as you do (as opposed to just strengthening their conviction that they're right and you're missing something), address the substance of what they're saying. Here are some responses to your claims. Pick a number, any number, and have at it! I may not be able to come back - I've already spent 15 minutes on this - but I'm sure others will respond.
1) "There are a lot of very casual accusations of inappropriate behaviour, so how true it is really depends on the individual case.... I don't see what that does to do with sex." See my statistics above explaining precisely what it has to do with sex, namely that men perpetrate 98% of sexual offences (and have up to 150% greater power with which to do so). I mean, 98%, Erin - 98%! You don't get much closer to an absolute than that in human behaviour. Did you miss this post?
2) "Men who are uncomfortable around gay men should have the right to kick them out of places too." Please tell me that you're not suggesting that gay men are 98% more likely than straight men to commit acts of sexual violence? That's an outdated, homophobic trope, but would be the only logical explanation for this analogy, given the stats above...?
3) "Trans people have been using the restrooms they wanted for decades." Exactly! Again with reference to the stats above, it's appalling that women were never consulted.
4) "Why was none of this being given any attention until recent years?" Previously, access was limited to a minority of old-school "transexuals", with the social contract (common understanding) clearly acknowledging that this excluded transvestites and "male-presenting" men. Now, in contrast, the exponential expansion of "trans-" because of self-ID, and efforts by individuals such as yourself to encode this in policy, law and the social contract has made it an exponentially greater issue. The real tragedy is that your campaign has negatively impacted on those old-school transsexuals, too.
5) "And if someone is behaving particularly anti socially, then they should be removed." Who by? The women there? Above, you argue that "You can't safeguard an open door" - in other words, you say that the previous social contract (which meant that any male walking through the female toilet door would be highly likely to be challenged by witnesses, including other males, outside it) is useless. Yet despite this, you appear to think that an intimidating or abusive male can simply be challenged/removed after he's begun to behave as such, by the few random women trapped inside with him?
6) "Do you want people who look like men to use the women's restroom?" A lot of us are pretty much OK with trans men using them, as most are recognisable as such (as you say, "look like men") and the stats show they're not a risk to us or our children, yes.
7) "You can't safeguard an open door" Cf. pubs with age-restricted entry, school buildings without security, the cushy head office at any company etc. People don't enter randomly, and no security is needed. How is "you can't stop them, though!" an argument for doing away with the common understanding that pre-teens / strangers / the basement minions of the business can't enter those spaces?! Society works because of "the social contract".
8) "That doesn't mean it's wrong or that everyone else needs to cater to it." You've got this the wrong way round. Common sense shows that if 1) single-sex and mixed sex are provided, everyone's happy. Whereas if only 2) mixed sex is provided, some women are excluded. Why do you argue for 2) and not 1), as we do. Our argument, 1), removes the "everyone needs" situation that concerns you so much.
9) "Public toilets aren't safe spaces. What we want doesn't mean they should be human rights issues." Woah. Are you actually arguing against the various campaigns for single sex loos in the developing world?https://borgenproject.org/connections-between-bathrooms-and-girls-education-in-africa/