I haven't understood the problem yet, Top. Here are the two typical types of bypass valve:
http://www.generalpump.com/PDFs/Unloaders.pdf

Neither valve is shown with a no-water safety shut-off of the engine, but it would not be difficult to add a pressure switch on the intake side of either of them, to actuate the engine's kill switch if there is no water-main pressure there.

The type of unloader I'm more or less familiar with is the first one, the Trapped Pressure type. That type of unloader just senses that the outlet pressure has become unrealistically high if the outlet is shut off. The high output pressure triggers the bypass valve, which reduces the pumping pressure to almost nothing, allowing the governor to reduce the engine's throttle opening to maintain governed speed at no load. When the pump pressure drops, water starts to flow backward from the output hose, and that causes the one-way valve on the output side of the valve to shut off, which traps maximum pressure in the hose and wand. If you have this type of valve you will probably notice that after the wand has been shut off, there is a brief surge of very high pressure from the nozzle when you first pull the trigger. After less than a second the pressure is down to its normal level, and the bypass valve closes, while the engine's governor opens its throttle to maintain the pressure and flow. Of course you will notice the large increase in the engine's exhaust noise.

Is that the type of unloader you have? If so, what is it doing that it shouldn't do?

The second type of unloader valve is the Flow Actuated type. It avoids having an over-pressure situation in the output hose and wand when flow is shut off, but it doesn't really unload the pump. It uses a pressure tapping and orifice in the output side of the valve to sense when there is no flow, and when this happens it opens the bypass passage as the other valve does, but it is only a small passage and it maintains full operating pressure from the pump. Hence with this type of valve you would not notice the engine dropping to no load when you stop spraying, and you would not notice an overpressure in the spray when you first pull the spray trigger. However because it is bypassing water at maximum pressure through a small orifice whenever you are not spraying, it would heat the water, which after a while would boil and cause cavitation damage to the pump. This type of valve therefore requires a thermal or time-delay shut-down feature to stop the engine if you let it run for any length of time without pulling the spray trigger.

While I haven't actually owned or used a washer with that type of valve, I'm not in any hurry to do so: it seems to me to be crude and also to be inconvenient for the user. Of course on a pump driven by an electric motor, it could be a good solution.

Do you have that type of valve? If so, what is it doing that it shouldn't do?