I've only been DSL in academies that are part of Multi Academy Trusts, three different large MATs. I don't have experience of LA run schools, so they may have different models.
In a MATs, especially the larger MATs, academies and their Heads have very little say over the content of key policies (like child protection, behaviour, code of conduct etc). They have a degree of control, because Headteachers know the context of their local community, but surprisingly little and for some policies they will have no say at all. Mostly Headteachers and SLT are paid to put in practice Trust policies. Smaller MATs may listen to the wishes of the Head more, but policies are still driven by Trust it just might be more of a collaboration rather than dictatorship.
Having said that, I've just read the Brighton & Hove Trans Toolkit (with my DSL hat on) and it states clearly that decisions on use of the toolkit sit only with the Headteacher. So regardless of the above, this toolkit isn't mandatory and so this Headteacher is deciding to do this, rather than bring made to.
If I was DSL, I would be pushing back to the Head with regards to this. But that said, I've previously worked for a different Local Authority who were very pro-trans. It was hard for our (MAT run) academy to challenge the rhetoric coming from Childrens Services. This was at a time when pushing on that was would be labled transpobe and biggotted. This is less the case now (except B&H, it seems).
Good luck with your mission OP.