Dusty,
I like that idea that you mentioned of dumping the preluber on startup, then shutting it down until the motor is up and running. I was always concerned that with the solenoid open when starting, the first quart of oil pumped would go back into the perluber and not the engine.
I would have to see that valve you describe as a one way valve or a schematic. In my mind it is actually just the solenoid valve. Many of these systems do come with a one way valve. If installed properly, it allows flow only to enter the engine, and not backflow back into the oil filter etc. where the pressure would be lost. One way valves are just that, one way. Most are mechanical like a spring against a ball. When pressure comes one direction against the ball it pushes the spring back and allows oil to flow. The other way the ball is just forced tighter against the seat not allowing any oil flow. The other kind I am familiar with is a flapper valve which is usually more expensive. Same idea just using a flap that pivots. One way pushes the flap open; the other just forces it tighter against the seat. Neither can be made to flow in the opposite direction.
My interpretation of the system you have is by pressing the button it is opening the solenoid valve allowing the higher pressure oil stored in the preluber to flow to the engine, which is at a lower pressure or zero for startup. As I said previously there could be a one way mechanical valve installed especially if the installation did not go directly to the block and tied it at the oil filter. Then when you release the button the solenoid closes which would not allow flow back into the preluber. Next, the ignition key is turned on which again opens the solenoid allowing the higher oil pressure to recharge your prelube bottle for 20 seconds, then the timer times out, closing the valve, preserving the pressure in the preluber. Hope this makes sense. I have to see a physician today and he will be carrying more than a stethoscope. If any of this is incoherent or offensive, I apologize. Had to take the pre visit meds.
With your stethoscope trick, it is probably good you don't install toilets.
Bill