When you install a 120V appliance on a 240V electric line, you are over-driving the heater element at a little less than 400% of its normal power. This is really hard on everything- the wires, the heater elements, the thermostat, etc. Depending on the element, it could trip the breaker on a typical 30A / 10ga water heater branch circuit.
I wonder if your original contractor therefore put a bigger breaker in that is inappropriate for the wires. They clearly didn't know what they were doing in at least one major regard (wrong voltage); I'd check to see if they didn't cut another corner with putting a 40A breaker on a wire rated for 30A. That could be a fire hazard.
You can DIY this, it says on the front of the beaker switch what the current rating is.
Regarding putting the 240V-rated thermostat and heater element in, this is fine. It will fix the problem by lowering the power from 400% to 100% of the unit's rated level, which is exactly what you want. A different, equivalent fix might have been to just change the line in the electric box to make the circuit run at 120V. But course the plumber can't do the electric box since he's not an electrician.