A check valve shouldn’t be that hard to fix or make try cleaning it out and reseat the ball.
We’ll probably a little emery cloth and slap some lipstick on it and it will be fine it’s not like it’s a pressure release valve.
Where does the valve mount ?
Could it be the pressure switch and not the check valve? I have this one if you want to give it a try.
Your motor pulley looks like it’s for a wider belt. The belt could be bottoming out rather than squeezing between the sides. That would explain the belt slipping. The compressor pulley should be fairly heavy so it acts as a flywheel. The only other thing I can think of is that you need a good electrical supply. A 240V supply might be best. My 1hp motor would occasionally trip the breaker.
I‘m not sure what you mean by unloading. If you mean reducing the pressure in the cylinder head after a stop the I believe it just bleeds off through the intake valve.
Sure enough there is a small bleed hole that I can feel a puff of air from on each compression stroke. It would be easy to duplicate if the part you ordered doesn’t have one.
Mine is a 2 cylinder and as mentioned before has a 1hp motor.