The issue is discussed, perhaps not completely clearly, in Henneaux and Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. They prove, in chapter two, that the Dirac bracket is precisely the Poisson bracket of $S$ determined by the symplectic form $\Omega|_S$ which is the pullback of $\Omega$ from $P$. In fact, this approach is preferable over the use of Dirac brackets, as I understand the story, because pulling back differential forms is a more elementary operation than generating these elaborate Dirack brackets.

The issue is discussed, perhaps not completely clearly, in Henneaux and Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. They prove, in chapter two, that the Dirac bracket is precisely the Poisson bracket of $S$ determined by the symplectic form $\Omega|_S$ which is the pullback of $\Omega$ from $P$. In fact, this approach is preferable over the use of Dirac brackets, as I understand the story, because pulling back differential forms is a more elementary operation than generating these elaborate Dirac brackets.