I have no idea of your systems, but once I had a car the ran pefect but died in a hard left turn, After several trips to the mechanic it turned out the carburetor float did not float. Changed the float and the problem disappeared.

I don't know it there is a float in your carburetor but could something similar happening.

I had week one on Briggs twin-( same pump ) last week. With the hose off it barely dribbled. But krnuttle came close when he brought up carburetor. I've seen water in the carb bowl do the same thing.

Thanks for the comments. I can get the OEM Kohler on amazon (?) for under $15 and since I have three tractors with the same pump, will get one JIC. Still don't see how a plastic housing with a diaphragm can be "weak" DOA, yes, but not weak. Maybe a tiny bit of crud in the check valve, more diaphragm material. For that price, not going to worry about it.

Make sure the pulse hose doesn't have leaks.

Here are more thoughts to consider. These pumps do not have great capacity and deliver very low pressure to the carburetor. The suction line may develop air leaks when cracks form in the outer rubber housing and air is drawn into the fuel line. Air leaks also develop around the clamp connections when the fuel line is older and less flexible. There may be some dirt on the pickup screen inside the fuel tank. When you are mowing up the hill, the pump needs to pull the fuel a greater height. If the pump flapper valve become less flexible, some fuel may not be moving forward to the carburetor. A tiny hole could be developing in the pump diaphragm. As mentioned, the pulse line requires inspection. There may be problems with the engine breather valve or torn dipstick o rings. In warmer temperatures, vapor lock may develop when these factors are present.

All input makes sense , especially when you add that with 1\3 tank would provide less head to the intake to the pump . Once you filled the tank you develop additional hydraulic head to assist a potentially weakened fuel pump .

Thanks for all input. Tractor is only three years old and all hoses seem fine, Bill, pulse line was the first thing I checked. Running fine now, maybe I've finally seen one get weak. :-) New pump should be here in a day or so.