If you have an RV fridge or electric water heater, you probably want those also not powered by the inverter.
Yes, as you realized, you do need a subpanel. The AC and DC sides, although mounted on the same outer chassis, are separated electrically and physically so are not really part of the same panel/enclosure. Besides the main breaker issue, you would need to isolate portions of the main bus, which would require unapproved modifications, and have two separate neutral busses. I think it is possible to have two separate sources in the same enclosure per the NEC with certain required labeling, but I'm not positive about that.
It might work out have the new panel be the main panel and the existing panel as the subpanel; if you need fewer breakers for the main, it might mean being able to use a smaller additional panel.