  The decay B -> J/Psi K_S remains the most important source of information for
the B_d mixing phase, determined by the CKM angle beta in the standard model.
When aiming at a precision appropriate for present and coming high luminosity
colliders, the corresponding hadronic matrix elements are a major obstacle, as
their precise calculation is still not feasible with existing methods. Flavour
symmetries offer a possibility to extract them from data, however again with
limited precision. In this article, we propose a framework to take subleading
contributions in B_{u,d,s} -> J/Psi P decays into account, P=(pi,K,(eta_8)),
using an SU(3) analysis, together with the leading corrections to the symmetry
limit. This allows for a model-independent extraction of the B_d mixing phase
adequate for coming high precision data, and additionally yields information on
possible New Physics contributions in these modes. We find the penguin-induced
correction to be small, |Delta S|<~0.01, a limit which can be improved with
coming data on CP asymmetries and branching ratios. Finally, the sensitivity on
the CKM angle gamma from these modes is critically examined, yielding a less
optimistic picture than previously envisaged.
