Question

Is my proof of Proca Lagrangian local gauge invariance correct?

Answer 1

My task was to prove that the first term of the Proca Lagrangian is invariant under local gauge transformations. ☺ ☺

Answer 2

to put the gauge transformation A→A+dλ(x)A→A+dλ(x) in the leftmost term, which would cancel with the second term as the function λλ is analytic in xx, so the derivatives commute. Your justification regarding linearity is not true as the only thing you need to assume about λλ is exactly that it's analytic (otherwise the derivatives don't necessarily commute), but otherwise completely arbitrary