Hi Tom,
These guitars are new to me as well. I've never seen a headstock quite like that for nylon.
For what it's worth, in all the years I've been playing and changing strings, this kind of thing has NEVER happened to me. Never had a string break while tuning them after putting them on. And I usually tune them about a half-tone or tone higher after just putting them on to give them an extra breaking in stretch before the guitar is put away.
You definitely should not be experiencing this at all.
Question: What is the nut width on this nylon guitar? It looks pretty slim.
I'm thinking that the angle between the nut slot and the post hole while you're tuning may be putting a lot of pressure on the nut and the tuning post, and that the nut is likely to be experiencing the most stress in the encounter.
I agree that this is the area of concern.
And the carbon trebles may indeed be more vulnerable to that kind of stress. Don't know for sure though. I don't use or like them.
A traditional headstock and nut width on a classical would lessen these forces by a lot, since the "break" angle between the nut and post is a lot less than what your headstock would allow. In fact, I always try to make it as straight a line as possible.