Welcome to the Energy Matters Forums Mrsbobby
If you have a suitable inverter and battery, there is no problem operating large loads in an off-grid system. I regularly use 2.4kW power tools- saws etc, MIG welder, microwave, induction cooktop, vacuum cleaner, dishwasher etc, sometimes several of the above at the same time. I currently have a 7kW inverter, but even with the 3kW inverter I had a few years ago, using the above appliances separately was no problem.
I only use a small generator for battery charging in cloudy weather, as I have air and water pumps that must run for aquaponics systems, which represent a significant amount of energy usage, almost as much as we use in the house, or significantly more in summer when I run a large (>2kW) water chiller.
You probably should think about getting a quote for Lithium battery rather than the Lead-acid, and perhaps for a larger PV array to reduce generator use in cloudy weather.