Do you have access to this?

Moureau, Berat, Pitsch (2007): An efficient semi-implicit compressible solver for large-eddy simulations. Journal of Computational Physics 226

There you will find useful references on the projection method for incompressible (costant and variable density) flows, plus you will see how the method is adapted to handle some acoustic features.

However, i repeat it, it is very unlikely that the pressure-velocity coupling is causing you troubles in handling shocks. Indeed, i suggest you to actually use your current method (however the PISO from the Jasak's lecture notes is probably a better option than the SIMPLE) and look for something else causing your problems.

Still, i realize from you posting the Jasak's notes, that the modifications needed by a segregated solver to handle compressible flows are new to you. So, i guess if they are also new to your senior... in that case there could really be a problem in your SIMPLE implementation.

a failure in capturing schock with good resolution does not necessarily depends on the failure of the algorithm for compressible flows... maybe your spatial (and temporal) discretization is simply not good enough for problem with discontinuity