I am no expert on DRASH. I avoided working on them when I could.
So, I will try and explain how the system works. The sensor, or better said, Fuel Pressure Switch, is a warning, that the filters are dirty or getting to a condition that is close to being plugged up. The FPS had a spring loaded switch that remains open as long as the output of the filters assembly is above a certain limit. When the output goes down too low, the switch opens up, reverses the contacts in K5 and shuts the engine off.
The reason that the set runs in battle short is because the battle short bypasses certain safety features. The same safety features that are bypassed when you turn on the set. On startup, you have no oil pressure. You have no fuel pressure. You have no AC output, and so on. K5 is one of the safety circuits that gets bypassed, until you let the start switch go. The battle short "jumps" all those safety. You are forcing the set to run in a condition that can possibly harm or cause problems.
So, if switching out the K5, proves that the relay is good. That leaves only the FPS and or restrictions in the fuel system.
One other possibility is:
Have you ever seen the set run right? If not, then comparing the wire diagram to the relay is not a bad idea. Someone before you may have tried to "repair" the set, and wired it up wrong.