And mixed sex hospital wards is probably the best approach to take in order to make ordinary people sit up and take notice.
If you focus on the mixed-sex aspect, then you don't even need to get into the transgender thing. You just have to state that women deserve same-sex accommodation on hospital wards. And the new NHS guidelines both promises this and denies it in exactly the same document. That's the issue.
Keep it simple. Keep it to the one issue.