The starter has nothing to do with the back firing. Several years ago, I owned a Grasshopper with Kohler 20 hp. The engine gave me problems at start up. The pro models now has seemed to remedy the problem with the installation of a electronic bowl float shut off. All engines pretty much comes equipped with them now. The other thing was that sudden flow of fuel into the carb and then the shut off. The excess fuel burns, pops through those large muffler systems.
I would make sure your carb is clean, check fuel filter. I might ask if this mower is equipped with the loop fuel system. If something is wrong with the air flow.....the epa thing, the mower will act funny until it is replaced. Anyway, when you fill up the tank, does the gas cap make a ratchetting sound. Similar to your car cap, if it is getting too much air into the system, things go wrong.
I can't help much more than these little tips. I really don't think it is your starter.
Take the thing off and take it to O'Reilley's, Auto Zone, or Pep Boys and have it checked out before you invest in another one.