interesting problem.
it sounds like you could make things work without the battery and inverter. sure you could go this route, but batteries are heavy, don't last forever, make a mess if acid gets out of them, and would have to be wired and de-wired each time they were set up. another thing about batteries, is if you draw them close to dead on a regular basis, they won't last long. a general rule to make batts last a reasonable amount of time, is to never draw more than 50% from them. drawing even less per use will make them last even longer.
i'm thinking of some sort of a timer, relay or programmed system where your loads are prioritized. IE: a system where only one large load is turned on at a time. imagine a clone of yourself constantly plugging in and unplugging loads at a fairly fast rate to keep the temps at the sweet spot per appliance while keeping the overall draw at less than 10A.
i'm far from the right person to design or build the unit that could accomplish the above idea, but i know one could be made. it sounds like you have enough power if you could send it to exactly the right place at exactly the right time for exactly the correct amount of time.
another option would be to use all 10A to charge a smallish batt bank with an inverter attached. the problem here is there are significant losses both in making DC from AC, as well as turning the newly produced DC back into AC. i'm guessing these losses wouldn't outweigh the gains of having a battery "buffer" for the times where all the large loads need juice at the same time.
adam