In your derivation of the EOM, you got some indices wrong and forgot an A^μ. Maybe there are also sign or coefficient errors, haven't checked it in detail. I propose you to redo it in more detail.
For the next step, derive your EOM once and calculate ( θ η^{μν} - ε^{μνα} ∂_α ) J_ν using your EOM and the derived version, which should give you the left side.