I think they are equaivalent, at least on the constraint surface, since for the physical sector this is so (see DeWitt, Global Approach..., Vol I). However, you should remember that these brackets are defined on the different spaces: the Peierls bracket is defined on the configuration space, while the Dirac bracket is for the phase-space.

As for the gauge dependent quantities, it is does not matter. You can substitute to these brackents any function on the corresponding spaces.