Sounds like your battery isn't supplying the required current to start the genny. The 13.5v you measured is either the converter voltage if you were plugged in or the surface charge on the battery. A fully charged battery, after the surface charge has leaked off, is 12.6 or 12.7 volts.
Even if your battery is fully charged it may not have the capacity to supply the genny starting current.
If the battery voltage when trying to start the genny is low, you may need a new battery.
You might want to check the water level in the battery too.
