= Solution
In the flattened coordinates, $\partial_x=\partial_{\tilde x}$ and $\partial_y=2\tilde y\partial_{\tilde x}+\partial_{\tilde y}$. The transformed <formal adjoint> consequently has coefficients obtained from finitely many <real analytic> coefficient pullbacks and <polynomial> factors. They have uniform analyticity bounds on a fixed small ball. By the preceding non-characteristic argument, its coefficient of $\partial_{\tilde x}^m$ has modulus bounded away from zero there, uniformly for small $\varepsilon$. Division by it preserves uniform <real analytic> bounds.
The transformed forcing $\tilde g_\varepsilon$ has analogous bounds, with constants that may depend on $g$. Apply the <Cauchy-Kovalevskaya theorem> in normal form with zero normal data through order $m-1$ at $\tilde x=0$. The supplied <majorant series> radius result yields a common solution radius $r_g>0$ for all sufficiently small parameters. Tangential differentiation of zero normal data shows that the entire <boundary jet> through order $m-1$ is zero; changing coordinates preserves this vanishing.
For $(x,y)$ in the closed cap, $|\tilde x|\le\varepsilon$ and $|\tilde y|\le\sqrt\varepsilon$. Choose $\varepsilon$ so small that $\sqrt{\varepsilon^2+\varepsilon}<r_g$, with a strict margin. The whole closed cap then lies in the common <real analytic> solution neighborhood. Pulling the solution back proves
$$
\boxed{P^*v=g\text{ throughout }\omega_\varepsilon,\quad j^{m-1}v|_{\tilde\Sigma_\varepsilon}=0.}
$$
In fact $v$ is <real analytic> on a neighborhood of its closure, which is useful for the boundary integration below.
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