"What Posie may or may not believe is irrelevant"
No maybe about it. She been very clear about it multiple times, and it does matter, just as the motivations and beliefs of each of us matters, but especially if we wish others to follow our lead or respect or trust our versions of events.
"....the ludicrous position we are in now, where the OED definition of 'woman' is somehow deemed to be offensive."
It's not. The poster company were explicit that it wasn't the motivation behind the poster that led to its removal.
"That is a farcical situation, which is quite frankly abusive towards women."
It seems very wrong to me to devalue abuse towards women by putting this situation into that category.
"Those suggesting that anybody who believes in standing up for women's rights is automatically transphobic need to go back to school to learn to think critically."
I don't perceive Posie as standing up for women's rights. Even if I did, it's not that part that makes her a transphobe.
She's not automatically transphobic, but she is demonstrably, repeatedly, deliberately and persistently transphobic in her views and actions.