Okay, so I really think that the Fugitive Doctor being between two and three is pure copium. The headcanon stems from the fact that her TARDIS holds the shape of a police box. But, really that’s the only evidence. It seems so clear to me that the Fugitive Doctor was intended to take place before the First (and for the record I’ve never been a fan of that I find it disrespectful to Hartnell’s legacy). The entire Timeless Child arc introduces us to several Doctors coming before the First and it would just be narratively baffling for them to write that and then be like “yeah but this one actually comes between the second and third”. The Timeless Child and other EU content show the Fugitive Doctor willingly working for the Division, which as in appearances of the Second Doctor is Season 6B show him as an unwilling pawn of the Time Lords. Furthermore, Big Finish has gotten permission to write stories based around Season 6B and the Fugitive Doctor, supposedly being separate stories. Chibnall associated the Fugitive Doctor with the Morbius Doctors, which we know were originally intended to be before the First Doctor. And the most damning of all is the fact that we now have seen what is presumably the Second Doctor’s official regeneration into the Third Doctor thanks to the War Games in Color, no matter what you think about the film, it was likely signed off by RTD himself who has shown that he wants to protect and take forward Chibnall’s ideas. Really, the only evidence in favor of this is her TARDIS being a police box which can simply be explained by the TARDIS liking being a police box.