His intent was to pay a professional to solve the problem. If the new pump had solved the problem, he would not have stopped payment on the check.
My calculations show that since the 12 GPM, ½ HP pump will build to 70 PSI and shut off, the water level in the well can’t be any lower than 109'. Pumping from 109', and holding 29 PSI, that pump should be delivering 11 GPM. You can add up the amount coming from all the open taps and check that out. Even if the water level pulls down to 160', that pump should deliver 7 GPM at 30 PSI. The problem is that holding 30 PSI in the basement, is only going to deliver 16 PSI to the upper floor.
If you had measured the GPM while running at 29 PSI, I could have verified the pumping level in the well. You could also run just the upstairs shower by itself, measure the GPM in a bucket, and check the pressure while you are doing this. If the pressure is holding steady, I can check the pumping level of the well from this as well, and you can check to see if this is the kind of pressure you want to see on the upper floor.
I would want a pump that could deliver about 10 GPM at 60 PSI in the basement, so I still had 46 PSI on the upper floor. A 3/4 HP, 12 GPM pump could deliver 9 GPM at 60 PSI, even if the water level in the well is 160'. Then it would also deliver 12 GPM at 60 PSI, if the water level never dropped below 100', which I think is the actual pumping level.
I believe the ½ HP is just not large enough to deliver the flow and pressure you need. If I were the pump installer, and you proved this to me, I would change out the ½ HP pump for a 3/4 HP, and only charge you the difference between the two pumps.