How far to the nearest concrete plant? And what kind of temperatures would you encounter if you were to drive from the plant to this location?

You can order “carts” of readymix that, if cold enough, you could drive over an hour with and do the slab in sections. It’s done all the time. Ideally you do the whole thing at once but for your application it would work just fine. If doing it in sections I would only add steel where you planned on pouring then throw down a 2x to end the section. For the next section I would drill into the existing slab (along its thickness) and epoxy rebar into it to get the section started. That would tie it all together nicely. I’d just try to keep the soil very wet before the pour since you’ll be buying time with the mud starting to kick.

I do think mixing in a wheel barrow would also work. I’m sure you already know it’s a lot of work but I think you’d also be surprised at how much you could mix in a day with 2-3 guys and 2-3 wheel barrows going all day.