Problems could well be unconnected but I would find the cause of the bleed down before doing anything more to address the stumbling.

Fuel pressure is controlled by fuel pressure regulator which would not be in the mix when the pump is off.

Yes, running rich can shorten life of cats but it would take a very rich condition for a long period of time. Cats get very hot and would burn off the majority of unburned fuel.

If your fuel pressure stays normal while engine is running for 5-10 minutes pump is working normally. I would block the return line, turn the key on to get pressure to normal and then turn the key off to see if it holds pressure. If it does that means the problem is on the other side.

Might be worthwhile to get a propane tank and the heater element that attaches. They put out an amazing amount of heat. Not expensive.

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X5 4.6 2002 Black Sap, Black interior. 2013 X5M Melbourne Red, Bamboo interior
Dallas