The Honda non-inverter generator should be just fine. Many of the Hondas have a built-in Automatic Voltage Regulator (AVR) which is a step up from a basic non-inverter generator.
The way these non-inverter generators can cause harm is when you let them run out of gas while things are plugged into it. The engine will surge up and down, which causes the power output and frequency to surge and droop as well. It is best to remove the load and shutdown the generator yourself before it runs out of gas to prevent any issues. Like I said, Honda generators are pretty bullet proof. They are less likely to have an issue with power surges with an AVR, but the power will droop (brown-out) when the engine is running out of gas.
I have one of each. A little Honda EU2000i inverter for camping and an Honda EP2500CX (with an AVR). Both generators power my fridge, furnace, TV, sump pump and internet quite well during power outages. The non-inverter generator handles surge loads better (like when the furnace blower or sump pump kicks on) because it has a larger engine, so I favor it when it's cold outside. Never a problem with either one, I use them both without worry.