Not sure about the cutoff issue. All tankless on demand units require minimum flow rate to kick on - that’s the down side. You could put a small well insulated tank down stream of it, closer to your water faucets to get hw quicker and in small amounts.
A honda variable speed -2kw generator would run ~ $1200 jumping up from there you are looking at close to twice the price
The most elegant working RV retrofit solution for on demand tankless hot water that I’ve encountered does involve two very small, well-insulated reserve tanks, one ahead of the unit to maintain flow and one behind the unit for reserve hot water, specifically for on-off “Navy showers.” The other, less elegant solution puts a hot water reserve at the shower head. Works well but looks a little clunky to me.