What kind of diverter is it supposed to be; in the valve or at the faucet?
What model control valve is it?
Faucet diverters can give you a shower when in tub position if the valve is mounted too high in the wall, or if the line to the tub is restricted. They depend on gravity to keep the water from going out the shower when the tub is selected.
I suppose it is possible that they plumbed it with the tub outlet going to the shower and shower outlet going to the tub. I don't think you can fix that by changing the cartridge.
You should be able to remove the cartridge from the face of the control. It is only a bit over an inch in diameter. It is usually held in with a clip.