I don't see anything that definitively states what the actual oil pump bypass pressure is, but here are the normal operating pressures:
It's fairly safe to assume the system pump pressure relief valve cracking pressure is around 110 psi.
Corey: You did get an Alldata account, right? If so, the above info is in the engine specification tables. This info is for the 6.4L, but it only shows overhaul/inspection tolerances for 5.7 oil pump, so they are most likely the same.
As for oil filter pressure relief valve? Well, it doesn't technically have one, per se. Not one that effects system pressure, anyway. For the most part, bearing tolerances create the resistance to flow, which dictates indicated system pressure. The oil filter is in it's own little microcosm down there. As eluded to by others earlier, it actually has a PSID (differential pressure) relief valve. All it sees is the pressure difference across the filter and that is what it reacts to. It's definitely possible that the two can cross in certain conditions, but it's tertiary and coincidental.
There are many good threads on this subject out there if you care to go real deep into the weeds. For all intents and purposes of the question asked, it is possible that the viscosity or contamination can cause the filter to go into a bypassing state, but very highly unlikely that is what you are hearing, as it's not an on/off affair; it would just quietly bypass and you'd never know it. Without having gauges installed before and after the filter, we'd never be able to tell if it were bypassing, so the pressure at which it is set to bypass is really irrelevant.
For you aircraft guys... that's why there are DPIs, right?! When a DPI pops, what happens to system pressure? Nothing... unless there is a problem with the pump or other component in the system. Same thing happens here, but it always resets itself by spring tension and there is no visual indication. It's just a very very simple hydraulic system.