Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback of $C'$ to $K$ is birational to $C$?

The answer is obviously no, in fact I think it's obvious that any curve can't be defined over a field smaller than its field of moduli in the moduli space of curves. But this isn't true. Take the curve over $\mathbb F_p(t)$ ($p\neq 3$) defined by:

$y^p-y = x^3+tx$

Then that curve is birational to the curve defined by:

$y^p-y =x^{3 p^n} + t^{p^n} x^{p^n}$

which is defined over $\mathbb F_p(t^{p^n})$, because, by Artin-Schreier theory, they both represent the same cover of $\mathbb A^1$. But it's easy to see that the curve over $\mathbb F_p(t)$ corresponds to a non-isotrivial family of smooth curves and hence is defined in the moduli space of curves by a finite degree subfield of $\mathbb F_p(t)$. (The way I see it is by observing that the monodromy of the cohomology sheaf of this family of curves isn't finite.)

Of course the problem is that the moduli space of smooth curves is a Deligne-Mumford stack but the moduli space of regular curves is not.

So does my intuition fail for every curve, or just some of them?