They are wrong on the law. The Equalities Act prohibits spaces and services that discriminate against one or other sex, but then gives an exemption which allows for such single sex spaces/services to be provided in certain defined circumstance. It is this exemption which allows for single sex toilets, changing rooms etc to exist and not fall foul of the basic anti-discrimination premis of the Act.
The Act says nothing about those with the protected characteristic of gender reassignment being regarded as the sex to which they are transitioning. For that you have to have a GRC (though, even then guidance to the Equalities Act envisages situations where you might exclude someone of the opposite sex even if they are the same legal sex).
If I remember correctly, there is no case law which has decided that the single sex exemption is effective even if someone of the opposite (legal) sex is using that service/space. There is one first instance case (which does not establish law) in which a judge decided that a transwoman should not have been prevented from using the toilets at a pub provided for the opposite sex when a woman complained about the transwoman’s presence. I do not know why that decision was made (and, in any event, it does not set legal precedent), though I would love to know! I am not aware of any other cases on this - let alone those that impact the law - but maybe your councillor is better informed than I am?
IIRC, there are de minimis provisions built into the law so that a space/service benefitting from the single sex exemption doesn’t lose the benefit of the protection should someone of the opposite sex incidentally use the service/space, so I think this is what could be relied upon if a women’s refuge, for example, occasionally allowed a transwoman to stay.
However, Stonewall law has it that it is OK to provide a service/space for women and men that self-identify as women. So far as the Equalities Act is concerned, I do not see how that can fall within the single sex exemption and therefore there would surely be grounds for complaint by men (who do not ID as women) that it is discriminatory to exclude them. In other words, the for the single sex exemption to be effective, it has to benefit a single sex class, not all of one and some (but not all) of the other.