uncommon questions - choosing a frame and varying

Covariant electrodynamics defines a Lagrange density \(\mathcal{L} = 1/4\:F_{\mu\nu}F^{\mu\nu} + 4\pi/c\:A_\mu\jmath^\mu\), which is Lorentz- and gauge-invariant, leading to a Lorentz-covariant and charge-conserving field equation through variation. Would it be possible to fix the gauge and/or assume a Lorentz-frame before the variation and still obtain the field equations, only for the fields \(\vec{E}\) and \(\vec{B}\) directly?