Modulating a (diaphragm/direct lift) solenoid valve for proportional output
What you're looking for is called an "electronic proportional valve". And they cost at least 10 times what you are willing to pay.

Diaphragm valves are intrinsically "snap action". Once they get near to complete closure, the water pressure forces the valve completely closed. Proportional valves have no such bias, and as a result are not normally open or normally closed.

Sorry.