I have a Honda EU2000i which I used for exactly that purpose (until I installed solar panels). As mentioned, its DC output is limited, we get a much higher/faster charge using the AC output to run our Freedom2500 inverter
One "nit" is that I had to reduce the "power sharing" setting of the Freedom charger slightly to prevent its initial bulk charge from tripping the Honda's overcurrent limit. On paper and as measured with an ammeter the charger's AC draw is still just within the generator's capacity, but in the first 20-30 seconds or so the charger often trips it. Reducing the charger to (if I recall) some 80% of max lets the Honda survive the Freedom's start-up surge and hasn't materially affected my charging on shore power (come to think of it, I don't believe I ever set the charger back to "max" since changing to solar).
The Honda is pretty quiet at idle and under load (obviously not as quiet as solar panels, but a lot smaller than three 130W Kyocera panels and the arch that holds them up). ECO mode reduces the noise under light load and automatically throttles up on demand, so I'm not aware of any power reduction when using ECO, except that they recommend ECO be switched off with intermittent loads and to cope with heavy start-up loads. In fact, the sound of the generator slowing to near-idle aboard was an indication that the charger had backed off from "bulk" to "float" mode.
Interestingly, this thread prompted me to check the generator and Honda docs regarding the parallel option (and make three trips to my garage to check the serial number), and look for the (non-existent) parallel jacks. Short version: there is indeed an option to parallel two identical generators to increase capacity, but it is not present on my Canadian-purchased unit (presumeably due to its meeting stricter CSA rules vs UL). Looking at the schematic in the US version's manual, the parallel jacks are simply in parallel with the inverter
output, but before the circuit breaker. Personally I would not run two generators simply from the noise/space/bother aspect. While one is quiet, two not-quite-in-sync are likely to be less pleasant to listen to, even if theoretically only 3dB louder. Would be interested in feedback from anyone who has actually paralleled two of these units.
Still happy with the generator, now keep it at home to deal with occasional power failures.