The point of winding a transformer is to minimize the air space. So try to fill it up with copper. Litzing it will not do that unless you use a very small wire.
No bad Battery. I would put 2-4 good batteries of the same type you would use in the serial portion. This should leave you with some charged batteries if the serial drop to much. Even with one batt it should create a slight resonant's across the windings and the load that will help in the long run.
This thing will not run a huge load either. Thats not the point. We just want to run a small moderate load and maintain charge in the primary batteries for as long as we can. If your batteries are 100 amp hour you will not run more than 5 amp in the load portion.
It will be somewhat of a balancing act. The primary batteries will drop some and hold. If they continue to drop you can lighten the load or adjust the frequency. As the frequencies get charted we will start to increase the load.
If it rains tomorrow I will have time to get mine together (All I need is together just need a little wiring) to start exploring frequency issues. I am hoping I can cover the frequencies needed with the stamp but in case We might have to go with a faster solution. Not sure yet.
It will be a neat thing though have no doubt. I have already seen what this thing will do in pieces of other circuits.