Just to add. The Equalities Officer should only be operating within existing Union policy.
There are a couple of things in the OP that strike me as being potentially outside of the Officer's brief. Her/his role should be Branch member led, and in accordance with National policy. They just can't make up stuff on the hoof. So for starters.
• I’m currently writing a letter to Wes Streeting challenging the puberty blockers ban and the awful state of trans healthcare
I don't think this is National/ Regional/ Branch/policy is it? Has the Branch debated and agreed to this being done? I doubt that the membership are aware of it.
• I’m passing a motion to donate £200 to Trans Kids Deserve Better following their recent protests outside of the DfE and the LGB Alliance conference.
Individuals don't pass motions, even if a Union Officer. It's a collective decision by the Branch members. The Officer can draft a motion, but it needs proposing and seconding, then putting before a Branch meeting. It needs debating and then to be voted upon. Individuals can't decide to send Members money willy nilly to organisations that the Union has not voted to endorse. Have a word with the Branch or Regional Treasurer on this.
• We’ve worked with other local groups such as Trans Liberation Front (TLF) to mobilise an effective counter protest to Posie Parker and other transphobic events in our city.
As above. Have the Members agreed to this?
Union Officers should work within their brief and according to existing policy or as mandated by Members according to union policy. They can't go maverick and make it up as they go along.
Copy that directly to Sharon Graham with your comments as an official complaint. She will know exactly what is acceptable and what isn't. She's no fool. It would appear that the Equalities Officer is off on a frolic of her/his own.