(Turning two comments into an answer)

The definition of $L(x,\alpha+1)$ was wrong, instead it should have been $$L(x,\alpha+1)\leftrightarrow\bigvee_{n\in\omega}\bigvee_{\varphi}\exists p_1,\ldots,p_n \, \forall y(y\in x\leftrightarrow (L(y,\alpha)\land\varphi(y,p_1,\ldots,p_n)))$$, and as this is not $\Sigma_1$, $\Sigma$-recursion is not applicable with the corrected definition.

(Turning some comments into an answer)

The definition of $L(x,\alpha+1)$ was wrong, instead it should have been $$L(x,\alpha+1)\leftrightarrow\bigvee_{n\in\omega}\bigvee_{\varphi}\exists p_1,\ldots,p_n \, \forall y(y\in x\leftrightarrow (L(y,\alpha)\land\varphi(y,p_1,\ldots,p_n)))$$, and as this is not $\Sigma_1$, $\Sigma$-recursion is not applicable with the corrected definition.

Additionally, from step #1 on, the argument is dependent on stating an axiom of foundation in $\mathcal L_{\omega_1\omega}$ such that $M$ satisfying the axiom means that even just the ordinals of $M$ are externally well-ordered. But Karp ("Finite-Quantifier Equivalence", in The Theory of Models, J. Addison, L. Henkin, and A. Tarski (eds.), 1965) and Lopez-Escobar ("On Defining Well-Orderings, Fundamenta Mathematicae vol. 59, 1966) showed that external well-orderedness is not definable in $\mathcal L_{\omega_1\omega}$ nor in $\mathcal L_{\kappa\omega}$ for any cardinal $\kappa$.