The Duramax guys CP3's have a different primary gear that allows dump to pump and create a self priming condition and therefore can be run without a lift pump, cummins on the other hand dumps to line and doesn't self prime as well and requires the lift pump for assistance and after filter changes.
The D-max has a different pump, both in the gear rotor and other design pieces. It was built to draw fuel, unlike the Cummins version. The line size and location of fuel tank play a large part in the design also.
That pump has also gone thru multiple updates because the original version was garbage. A lot of those early design never went 10k before the complete shelled the fuel systems on them. The first 3 years or so of the Duramax engine GM warrantied the crap out of stuff because it just would not stay together, CP-3, injectors (largely due to FOD from the CP-3), and problems with the fuel control system that cracked injectors an dumped fuel directly into the oil. As long as the manufacturer steps up there is no hue and cry about problems so no publicity. My cousin had one of those nightmares, 3 times in less than 50k the fuel system ate itself up requiring complete replacement. Had is the operative word, he finally dumped that turd for a newer one. Didn't kill the fuel systems all the time so that was a step up. I just laughed at him, less than 50k and all the problems he had while mine was over 100k with a tuner for most of that and all I did was change the oil and filters. GM guys have no sense of humor. I personally know of 3 of those early ones that were nothing but a headache and it ALL due to the CP-3 killing itself. Why do you think LP's were such a benefit to those trucks? Aside from the 20 hp gain that is.
Low\no fuel to a CP-3 is just bad news.
The Cummins CP-3 version was NEVER intended to be operated without at least 3 psi of head pressure, that is straight from the Bosch shops on that. Lines too small, route too long and steep to draw fuel adequately. The generic CP-3 is rated from -5 to +20 psi operating range, but, that is totally dependent on platform. Yeah, you can run -5 psi on a gravity feed 1/2 line on a 6 ft drop but don't expect it to live in Dodge truck too long.
The design of the CP-3 prioritizes fuel to the rail, so no you would not see problems when the pump loses lube. The ECU will stretch duration to compensate and unless you have gauges and\or are using it hard there is no indication of problems.
The bulk of failures, contamination aside, is going to be from cavitation in the gear rotor pump. MUCH higher percentage of the D-max pumps suffer that failure. Basic hydraulics tells you that 5-6 psi is way low for gear pump head pressures. Head pressure should be at least 20% of output pressure (closer to 30% is better) if you want it to live a long happy life. I leave it to you to figure out what that should be with a case pressure of 75 psi.
The CP-3 is far from fragile like the VP pump was but it ain't bullet proof. It will take longer but it will not like lack of attention and when it destructs it likely takes injectors and rail with it. A $6000 repair for lack of a $120 gauge is pushing the odds.