I just wanted to throw this in the mix. This is not a comment on the OP's son in particular, who is in any event too young for this to be relevant, but on the broader picture. I'm trying to phrase this as sensitively as I can.
Most men are not sexual predators. Most men would pose no threat to anyone in the ladies toilet. But a small number are and do.
In the same way, most adults with conditions that impact on their ability to live independently are not sexual predators. Most pose no threat to anyone in the ladies.
But a small number do.
That may be for just the same reasons that some other men are at threat. Or it may be because their condition impacts on their ability to understand that some activities are private, and with consent only.
As an example, one teenager I knew had to move from a female optician to a male one because, once the hormones kicked in, he could not understand that he should not masturbate through the sight check. Why not, after all - it felt nice. Or that he shouldn't grab things he found appealing. He could not see a female doctor for the same reason.
Another couldn't be around a particular, common, female clothing item, because it lead to immediate arousal which he did not understand should not be immediately responded to.
Clearly this is a very difficult area, particularly as it is certainly not the individual's fault that they behave this way. It's part of the broader picture of their complex needs.
But some posts here have been very much, 'Of course he can go in the ladies once he's grown up - he's disabled, he's not a threat'. And, very sadly, that's not always the case. Not even with one to one support, as the OP is providing.
I have no answers. There should be better disabled toilet access so everyone's rights are protected. I appreciate that isn't always the case, which is wrong. I've advocated hard for a 'changing places' toilet in my work place, for just this reason. But I just wanted to flag up that it's an oversimplification to say that someone who has significant needs is therefore not a threat. They're probably not a threat, because most people aren't. But the two groups aren't mutually exclusive.
OP, given the places without disabled loos are also likely to be the ones who don't have large rooms with many cubicles, is one solution for you to ask a staff member to guard to the door to the ladies whilst you and he use it? So you have in effect private usage for the time you're there? Or at least that staff member could let women queueing know you and he were in there, so they could make their own decision on whether they were happy to enter.