I've read over 100 Doctor Who novels (perhaps not a point of pride

) and it's pretty rare and it's even rarer that he refers to himself as the Doctor in his internal thoughts- why would he? I don't refer to myself in the third person too often in my head.
Except we're not talking about a first-person internal monologue. We're talking about the literary device of writing a scene in the third person but telling it from within a particular character's point of view -- i.e. relating what they're thinking and what they perceive from their own perspective, and only describing other characters' actions from the viewpoint character's perspective.
For instance, from Boucher's
Corpse Marker, p. 21-22 of the newly released "Monster Collection" edition:
The Doctor had walked along the narrow gantry and peered through the observation ports of five more of the green-lit chambers before he was satisfied that what he was seeing was unlikely to be a coincidence or a trick of the light.
...
They didn't look to the Doctor like gill-breathers, so either this was a breathable liquid or these were different to the normal run of air-breathing bipedal clones.
...
It struck the Doctor that this might just be one of those deeply conformist societies whose disciplines involved a rigid dress code and whose repressions included a horror of nakedness. He hoped not. It was his experience that such societies tended to violence and Leela already had more than enough tendencies in that direction.
See? Third person, but within his thoughts. The standard way to write a third-person scene in the modern fiction style.
Another example is Alastair Reynolds's
Harvest of Time. The very first page I opened to at random, p. 97, starts with this:
The Doctor watched the Master's accommodation block sinking slowly back into the flooded pit, trailing the dark umbilicals of electrical, air and water-supply cables as it descended beneath the glowing waterline. Jo, anxious to be out of the enhanced radiation environment, had gone ahead of the Doctor. He did not much blame her for that, feeling no great inclination to linger himself.
When Virgin was publishing Doctor Who novels they banned writers from writing from the Doctor's perspective in order to keep mystery about him. Not sure if that mandate carried over to the BBC books, but a lot of authors followed it mandate or no.
The books I've quoted from are BBC books. However, I pulled out one of my Virgin books at random,
State of Change by Christopher Bulis, and here's what I found when I randomly opened to a page (p. 205-6):
Metaphorically, the Doctor was resting his mind as well; sitting back in a corner of his own consciousness with his feet up, as it were, and letting the personality of his third incarnation handle the physical side of things, and strike up a bonhomie with his fellow gladiators. It was hard to believe that men who might be dead the next day could still find time for humour and a strange sort of fellowship, but such was the case. Human beings never ceased to amaze him.
Of course, three books isn't that large a sample, but if Doctor's-POV scenes were that rare, then the probability of my randomly coming across them on the first page I opened to in two consecutive books would be extremely low. My ease in finding them suggests, therefore, that they are relatively common -- though it would require a much larger sample size to be sure.
(
State of Change is a Sixth Doctor novel, though. I can certainly buy that Virgin banned scenes from the Seventh Doctor's POV in their New Adventures line, since he was meant to be a man of mystery. But maybe that didn't apply to the Missing Adventures.)