Is the gauge transform field in electromagnetism a Lagrange multiplier?

So, we can say that as long as charge is locally conserved, the action is invariant under gauge transformations.

Yes, gauge-invariance implies that we can only consider$^1$ gauge fields in matter configurations $(\rho,{\bf J})$ that satisfy the electric charge continuity equation.

Can we not also say, though, that $\Lambda$ plays the role of a Lagrange multiplier that enforces charge conservation?

No, charge conservation follows from global gauge symmetry of the action via Noether's first theorem, cf. e.g. this Phys.SE post

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$^1$ There is a dual question of how to determine the matter configuration from a give gauge field configuration.