 Solution

The Cauchy-Kovalevskaya theorem cannot be applied to merely $C^2$, non-$C^3$ Cauchy data. It requires real analytic data.

There is also no $C^2$ solution of the Laplace equation on a neighborhood of a point where one of these prescribed traces fails to be $C^3$. By interior elliptic regularity, any such solution would be smooth and real analytic. On the real analytic hypersurface retained from the preceding part, both its restriction and its normal derivative would then be real analytic, hence $C^3$. This contradicts the prescribed trace. There is no solution on a neighborhood of all of $\Gamma$ with the stated non-$C^3$ data. This does not exclude solutions near other points where the data happen to be real analytic.