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We consider experimental constraints in the MeV region in order to determinethe parameter space for the $U(1)_X$ extension of the Standard Model, presentedin the first part of our work. In particular, we focus on the modelUV-completed by cold WIMPs. We conclude that the electron anomalous magneticmoment, the neutrino trident production and the relic abundance $\Omega_{CDM}$provide the most stringent bounds and, in particular cases, they are sufficientto exclude dark-photon ($A'$) models. By allowing the axial-vector coupling ofthe dark gauge boson $Z'$, the interference effect with the SM gauge bosons mayreduce the bounds coming from the trident neutrino production. At the sametime, such coupling allows a region of the parameter space already favored bothby the relic abundance and by the discrepancy between experimental result andtheoretical prediction for the muon anomalous magnetic moment. We emphasizethat light-$Z'$ interactions, non-universal for the two first lepton families,necessarily create a difference in the proton charge radius measured in theLamb shift of the $e$-hydrogen and $\mu$-hydrogen. Finally, we determine theeffects of the new gauge boson on the forward-backward asymmetry in $e^+ e^-\rightarrow \bar{f} f$, $f = \mu, \tau$, and on the leptonic decays $M\rightarrow j \nu_j l^+ l^-$, where $M = \pi, K, D, D_s, B$ and $j,l = e, \mu$.

Light Mediators in Anomaly Free $U(1)_X$ Models II - Constraints on Dark Gauge Bosons