There seems to be two main themes coming up on this thread: how do you define gender identity and how can you judge someone else's gender identity.
Defining gender identity
The Yogyakarta Principles define gender identity as:
"Gender identity is understood to refer to each person’s deeply felt internal and individual experience of gender, which may or may not correspond with the sex assigned at birth, including the personal sense of the body (which may involve, if freely chosen, modification of bodily appearance or function by medical, surgical or other means) and other expressions of gender, including dress, speech and mannerisms"
Further information on gender identity is on wikipedia en.wikipedia.org/wiki/Gender_identity
Reading these definitions it is quite clear that LM and also Sophie Cook have female gender identities. However, it is not possible to have a gender identity where you are a woman on Wednesdays but a man for the rest of the week. So the OP is not sincere when he claims to identify as a woman on Wednesdays. His subsequent comments to the media and on this thread reinforce his lack of sincerity.
Pippa Bunce's gender identity is bi-gendered which remains constant. However, her gender expression varies from day to day. She does not have a female identity so would not qualify for AWS or the role of women's officer in the Labour Party. She is also not able to change her legal gender under today's laws or under legal self id - and I doubt she would want to.
Deciding if someone's claimed gender identity is sincere
Criminal law is based on people's intent. For example, to be convicted of murder you need to have intended to kill or seriously injury someone. Defendants could argue that what they intended to do is entirely subjective, that they did not intend to harm the victim and you have no way of knowing otherwise. Yet people are convicted of murder all the time. Why? Because the court looks at all the evidence and makes a judgment on whether the defendant is telling the truth or not.
Similarly to change your legal gender today you must intend to live in your preferred gender until you die. A Gender Recognition Panel (a tribunal) decides whether you are sincere. When legal self id comes in, a court will decide if an individual has made a false statutory declaration.
In the case of the OP, it is not trans people or allies who will determine if he is sincere in his claim to have a female gender identity some of the time, it will be a Labour Party disciplinary committee.