Water – pipe size to the shower

This is a very common setup, and basically the situation in my house. What you need is called a shallow well jet pump, and I believe this is what you are seeing in the store. Your suspicion is correct - that local mechanic is wrong. Some notes:

1. Size the pump 1/2 or 3/4 hp based on the flow rates and demands you expect. There's insufficient details to provide an accurate recommendation here.
2. You will need a pressure tank ("bladder") connected to the line after the pump in order to even out line pressure and prevent the pump from turning on all the time. This will very greatly extend the life of the pump.
3. To keep the pump primed, you need a foot valve or check valve on the end of the line that is submersed in water.

Demand the manufacturer's instructions so you can review them. Then respectfully insist that the plumber (who you paid to install it correctly) return and make all needed corrections. It's not right for the plumber to expect you to do anything unless you agreed to up front.

He should be correcting the piping if necessary, he should be adjusting the temperature stops, he should be adjusting the cartridge, he should be ensuring the trim is oriented correctly.