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I have Outback inverters cabled together to make 8000 watts available from the batteries. I will run a variety of loads such as the refrigerator and outlets, TVs, coffee maker, etc, but my practice is if I spend a night without shore power I run the generator until I go to bed and I never run A/Cs from my inverter.
My Outbacks have a “silent mode” in which they charge the Li batteries (and AGM) and then stop charging until battery voltage drops to around 24.4 or 24.5. At that point they resume charging until fully charged meaning they show about 26.5 or thereabouts, and then go back to silent mode.
Based on my battery performance curves my Li as currently used are projected to be capable of 14,000 charging cycles. So far so good, especially since I do nothing with them.