Why stop with Season 6B?
Think about it. The stories normally associated with a Second Doctor who served as a secret, deniable agent of the Time Lords in between "The War Games" and "Spearhead From Space" feature a Patrick Troughton who looks considerably older than he did at the end of Season Six. (Admittedly, Steven Moffat chalks this up to "shorting out the time differential", and suggests that Time Lords revert back to their youthful state when no longer around their future selves, but he doesn't say that they can't look old because they're old.) He mentions his age as roughly 400-450 in "Tomb of the Cybermen", and Romana gives it as 759 in "The Ribos Operation", meaning that there's a 300+ year span of adventures in there (with plenty of room for gaps, especially in between "The Deadly Assassin" and "The Robots of Death", but the point is there's a lot of time to be accounted for.) And the Time Lords discuss a change of appearance, but they don't really present their actions at the Doctor's trial as a capital crime (or 1/13th of a capital crime.)
And when the Doctor regenerates, it is always into the form of a man in the prime of his life--of all the regenerating Doctors, only Pertwee was over 50 when he was cast in the part. And we don't see that regeneration. And according to fan lore, there's a gap in that space where missing adventures occurred. And the Third Doctor has plenty of suspicious gaps in his memories.
What if the Third Doctor, as well as the Second, operated for years as an agent of the Time Lords? After finally regenerating due to old age, he regenerated into a young, fit Jon Pertwee (who I for some reason picture as dressing like John Steed from 'The Avengers') who eventually, after a long period of service to the Celestial Intervention Agency, was finally sent to Earth to begin his official exile. This Season 6C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y and Z would also explain where the new Doctor picked up his driving skills, Venusian karate and aikido, and tattoo.
I propose that once the Past Doctor Adventures return, we...what? I can dream, can't I? I propose that once the Past Doctor Adventures return, we see the final adventure of the Troughton Doctor, and the first adventure of his young successor. What do you say, BBC?