Small Engine Troubleshooting Problems? Ask a Mechanic Now.
The buzzing noise you hear is due to low battery voltage getting to the solenoid.
You stated that you have 13.5VDC at the battery. That tells me that you have your coach plugged in, and are reading converter voltage.
If you have a master battery switch, it must be turned on for the genset to start.
If you do not have a battery switch, or it is already on, you likely have a bad battery. Take it in and have it load tested. Do not rely on a voltmeter or battery charger to tell you if the batter is good or not. Even a bad battery can read fully charged with 12V, but not have enough ampacity to do any work.