Bluntly it sounds like a bad design.
Sorry, that was rather blunt, but it's still a huge amount of excavation for a problem that doesn't need that as a solution - rather inelegant, I'd say.
Hole saws are good, but your design choices leave you without an easy solution to your problem of achieving a flat bottom to the hole. Most of the usual things, such as a router bit won't work because of the depth.
Have you access to a bandsaw (or Japanese Hassumne handsaws)? It's common practice to make bandsawn boxes by rip cutting the base and glueing it back on again afterwards. If cramped tightly, the glue line becomes almost indetectable, as you're fitting the wood back where it came from (minus a thin kerf, obviously), and glueing the same grain together, like repairing a split.
For what it's worth, the drills used to drive diamond hole saws for masonry and (I think) the things that plasterers use for driving stirring paddles have slipping clutches, for exactly the wrist safety problem you mention. My old Makita SDS drill is very good and has one too, although these days it slips more than it used to (doesn't actually matter in practice).
And... the wear on a Forstner bit used that way would be enormous. You should use it in a drill press. The fact you can get away with it handheld doesn't mean it's a good idea. They need decent pressure to cut properly, and without it the edges will overheat, lose temper and blunt quickly. At which point you need more pressure than before to even cut, and the whole thing gets rapidly worse. Also, on one that wide, unlike a holesaw, there will be a huge difference in cutting speed between the outer edges and the inside. It may leave a clean bottom to the hole, but I can't imagine it will be wonderful.
I'd cut the end/bottom off, use a holesaw, and glue it back together. When finished it won't show and you'll have decent sized lumps for the wood burner in the winter! Try it and see if you can get a nice join - I think you'll be pleasantly surprised.
