I don't quite understand everything you say there.the direct on line contactor should come with or you should fit the correct sized overload for the motor it's controlling. You should not have any need for that that thermal overload switch as well.
I'm assuming your current set up would start the compressor automatically as the pressure drops bellow as cetain threshold? This should be run through the controls for the starter so a simple DOL starter will not work as you require it.
You say the DOL Starter I'm getting should have a thermal overload switch?
this is the one I'm getting
So I could check that out.
The confusing bit you say is that the DOL will not work as I require. I run the switch for the motor from the pressure sensor, then that sensor is powered from the DOL. When the pressure drops, the pressure switch powers the motor. the pressure switch will be live as long as the DOL starter is switched on.