No generator, no propane.
I have used charcoal briquets for 25 yrs, maybe more.
i will admit that the old Hibachis were not great on a boat although the originals did a good job of cooking. Then they got really cheap and weren't worth the trouble.
i now use a Magma. Leave the briquets to burn out overnight so they are cold. Yes, I have checked. Leave the lid on and down so a wind doesn't blow ash about. Bag the ash and when we have access to a proper garbage dump the bag and contents go. Not overboard.
I use a light wide mouth, for ease of use , kitchen garbage bag to upend the Magma into and then store the ash bag in a heavier plastic bag. The one ash bag will usually contain up to 10 days worth of ash before I need use another.
The briquets are kept in Rubbermaid bins and the bins are out of the weather. They don't get wet and stack and store readily and neatly.
You can use briquets safely and no mess if you use a bit of imagination.
I also sometime buy some real charcoal, charred tree bits for some extra flavour. It is a bit more of a pain as some of the pieces can be awkward and too large. A Hatchet or a pry bar and hammer fixes that but I do that at home so the mess can be cleaned up.
Of course some marinas ban barbequeing but it is ALL bbqing, not just charcoal.
It can be done if you wish and as far as I can see it is little more trouble than dragging propane around unless you already have it.