It strikes me that effective opposition to changes would best focus on the proposed changes to operating public spaces and facilities. Rather than the trans issue.
A local authority will have a set of aims and objectives around encouraging use of facilities either to promote public health and wellbeing and/or to maximise revenue/minimise subsidy.
They will recognise a duty to keep users safe, and that if users feel unsafe they will stop using the facility.
They will recognise that some very odd, potentially unsafe, people use public spaces and facilities. There is already quite a body of knowledge within the management of public spaces and facilities on how to design or manage out potential problems.
Only yesterday at the Tooting Lido I saw a security guard (a way of managing out problems) stop a middle aged man on his own without a towel/bag who was trying to gain entry and politely turn him away. Reassuring as there were masses of small children running around.
The main problem with the proposals is the lack of safeguards. Essentially any old Tom Dick or Harry would be able to claim to identify as a women. Lovely and PC but without safeguards or a management and design review, other users may feel unsafe, indeed may be unsafe.
Is there a user/friends group for the womens pond. If not its not too hard, using leaflets and email to set one up and hold an initial meeting. Look at the Corporations aims and objectives, seek to work constructively with them to meet these. Then carry out a seperate short user and potential user survey aimed at identifying what users want and dont want. That way you can claim to represent user views and, effectively dodge the transgender debate.
You may expect users to articulate concerns about sharing communal changing space with 'self declared' women, genuine or not. But that is then something for the Corporations to manage. If they have any sense they decide their obligations are under the sex (not gender) discrimination act. And that the potential conflicts between different user groups (including dodgy blokes who are using any TG concessions for their own ends) is such that management wise it is easier/safer to steer natal men to the mixed pond.
My experience is that people running public facilities are keen to do a good job and like recognition/appreciation. They may not be able to express their concerns within their organisation (which is why gender equality people are running the survey) but may not be inhappy to have a user group who can articulate user concerns.
Sorry that was so long. I have done my share of campaigning to protect public facilities.