Can't think of a good reason why to not do that.
Keep in mind one very important thing. Be sure that genset is NOT powering a load when you go to do you fuel starvation shut down, else you will be soon purchasing a voltage regulator and/or a rotor.
Don't think it would matter much if you exercise it once a month or once every few months. When you do exercise it, warm it up before applying load, run it with a good load on it for 20 to 30 minutes, remove the load and allow it to run unloaded for a couple of minutes.
Always a good idea to run a fuel stabilizer, no matter what, try to keep the fuel turned over (use it and add fresh). Change oil annually.