The alternator on your engine is designed to bring the battery back up to 80% faster than the generator and converter. That said, the alternator on the engine can overheat if you don't run the engine at 1200 rpm or so. It just wasn't designed to recharge an almost dead battery at low rpm.
The suggestion to use your generator for two hours in the morning and two hours in the evening is a good one. That's what we've done. It won't bring your battery up to a full charge, but good enough. Adding a second battery is also a good suggestion.
