I ask myself if the demand of local gauge invariance - say $U(1)$ invariance in free Dirac theory -$$L_D=\bar{\Psi}(i\gamma^\mu\partial_\mu-m)\Psi$$ is enough to define the full Maxwell-Dirac-Lagrangian UNIQUELY? I read that adding the Maxwell term $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $ demands for other requirements like 1) simplicity, 2) Lorentz invariance and 3) renormalizability. Similarly you need those other requirements for the interaction term $L_{int}=-j_\mu A^\mu$. While simplicity is clearly a metaphysical requirement Lorentz invariance is a genuinely physical one. But I am not sure about the role of renormalizability here?
What is the deeper physical meaning of it and why is the demand for renormalizability physically like the demand for Lorentz invariance?