Practically all motorhomes can charge house batteries from engine alternator when working correctly, and I doubt yours is an exception. Whether alternator can keep batteries charged while using the inverter likely depends on what loads you are running at the time.
A 1,000-watt inverter, when running near full load, will require around 100 Amps of 12-Volt DC power. That would be tough to keep up over long periods of time, particularly since many battery isolation devices are not rated at 100 Amps or higher.
On the other hand if your inverter was only powering a TV and DVD, or a refrigerator, where it may only need 200 watts or less, then it's likely the alternator can make that up.
Running the generator normally powers the converter which also charges the batteries. As with the engine-driven alternator, there is a current limit that the converter is designed for. Many motorhome converters are rated below 100 Amps, so it too may not keep up with current requirement of the inverter at full 1,000-watt load. Fortunately, it shouldn't have to because once the generator is started, the 115-Volt AC power should idle the inverter (assuming it has transfer switch around it), so it no longer pulls from battery and 12-Volt DC system. At same time excess generator capacity will recharge batteries via converter.