I think I would be tempted to say something like, "If a girl was thinking of breaking up with her boyfriend and he threatened to kill himself if she did, should she stay with him for that reason? If she broke up with him and he actually did kill himself, would it be her fault for not staying in a relationship that was making her unhappy? To what extent is she responsible for his mental wellbeing?"
If you get to the point where your son agrees that it isn't fair for this girl to have to stay with her boyfriend because he has threatened to kill himself if she doesn't, then ask the same question about trans people.
What should the rest of society be doing to stop trans people from killing themselves?
Better mental health support? Absolutely, yes.
Women and girls accepting all their spaces being made mixed sex because otherwise some trans women and trans girls might be upset? No. Why is that their responsibility?
I'd also bring up the correlation vs causation point another PP referred to. If there's a correlation between being trans and having poor mental health, then arguably the rates of suicide or attempted suicide or self harm among trans people need to be compared against the rates among people suffering from conditions such as depression, not against perfectly healthy, happy people.
Either way, the answer is better mental health support.
If a man said he was suffering from clinical depression but that getting changed next to women makes him feel better, we wouldn't allow him to use women's changing rooms on that basis, would we? We wouldn't consider that it was women's responsibility to let him see their boobs so that he could feel better. So why is it women's responsibility to make trans women feel better by sharing their single sex spaces with them?