I was kind of afraid that was the pump. As far as I can tell, it's a plain centrifugal pump, not capable of developing high pressure. From the performance table, that pump will never give the sprinklers as much as 40 psi
These pumps were okay for sprinklers that can work at lower pressures, but that is not the kind of sprinklers that get installed today.
Assuming the existing pump plumbing is sound, all that's left would be a pump upgrade. A jet pump would be an obvious choice, as they can give as much as half again the pressure you get now from that Flotec.
One concern, though, is how much water the existing well can supply. Just because the water in the well is within ten feet of the surface when you checked it with the pump off, doesn't mean the level doesn't drop when the pump is on. If that level drops another ten feet, you have problems for any standard pump.