I don't think the car has cooling system diagnostics as sophisticated as that. I don't think it has any cooling system diagnostics other than your eyes and the temp gauge.
It sure sounds like a bad ground, but I'd start by replacing, not just inspecting but replacing all the fuses if you haven't done so in the last 10 years. You're looking at a $5 investment if you get fuses from FastLane. Next is to loosen, clean and tighten every ground point under the dash. You might have to pull the gauge cluster for this but that's an easy task with your telescoping steering wheel.
If I had the original head on my SDL, it would scare me to have an inaccurate temp gauge. These heads are prone to cracking when overheated. If the head's been replaced in the last 5 years or so, you probably have the tougher 4th or 5th redesign which is more resilient to overheating. Not impervious to overheating, just tougher than the original head.
It's easy to test the water pump. There's a weep hole to tell you if it's going bad. You can also loosen the belt and spin the water pump pulley by hand. Be sensitive to noise and roughness. There should be neither. Again, cooling system service is not something to postpne on in a 300SDL.