I thought the story was that she said that he admitted to her.
JMH, it’s possible that the letter was unknown until a certain point. Also, the Chazara would protect the Rambam but not his Sefarim. The Gra didn’t embrace this story because he couldn’t be sure of it. But I think your Kasha from Reb Avraham Ben Harambam is strong one. If anyone saw the letter it should have been him. (Maybe an Askan produced the letter — in order to make a Kiddush Hashem)
Chazi Man Gavra Rabba Ka Mashid Alei only works when it is the Sahadusa of the Gavra Rabba. When you are talking about something that the Ri Irgis or Abrabanel couldn’t have verified it doesn’t add that much validity to the story.
Sam, he might have voiced opposition to a certain Sefer but that doesn’t say anything about the study of Darkei Hashem. Sure, a discussion of arms and legs, faces and brains, doesn’t sound so metaphysical. But do you think the Ramchal has anything that violates the Rambam’s Ikkrim?
Besides, as I alluded to in the recent past, when your life’s ambition is to introduce and spread a certain idea, you won’t waste time entertaining every devils advocacy. it was important for the Rambam to demolish any semblance to corporeality and he did so sometimes at the expense of explaining Medrashim and other Sefarim. Think of how the Satmar Rebbe would knock off certain quotes from previous Rebbes, in order not to inhibit his Shitta. We have often in the Gemara where an Amora would quote a Mishna in a different Tanna’s name so that the Mishna should be accepted.
Let’s not forget that before the Ramban Kabbala was very quietly learned. The Ramban himself only hinted to Gilgul while the next generation spoke openly about it.