OP if you are a long time supporter of Liberty and share their values, then perhaps you should reassess your views on trans issues.
MN is a bubble of GC feminists with an axe to grind. They have an ideology to push (trans exclusionary 2nd wave feminism). There is also a lot of ignorance of the law or even worse people who understand the legal issues but deliberately mislead to achieve their own ideological ends. The chances of getting an accurate and unbiased view of the issues is minimal.
Liberty specialises in protecting human rights and is very much a legal organisation. Its statement "If you aren’t trans, updating this Act will barely affect you at all." is likely to have come from rigorous legal analysis of the proposals - not from ideology. It also match my own analysis.