@ Luque, the first thing I saw was ~20W of power, that is quite low to lift something of that size/weight, hence my comments of increasing power. My quad is 430g all up weight, makes 9:1 thrust and pulls a peak of 120A 1800W on 4S-1300 LiPo.
I was thinking bi-filer to pull more current and H-bridge to turn in into a 1 stroke instead of 2 stroke. And yes, thin wire with high turn count is more efficient. One issue is high turn count is high inductance and we know it takes time for current to ramp up, something to think about.
Last year I was trying some different ideas, I wanted the current to rise faster, so it was something like 16 turn 10uH coil at 45A pulses, so i came up with a half turn coil (basically a short), 30nH 200uΩ (300x less inductance) and was going for ~700A to create similar flux, I was getting 700A pulses, but only a few till my supply could not refill the caps fast enough. Since then I discovered used server power supplies on ebay, for $15~$20 you get a fixed 12V at ~70A and they are tiny, great bang for buck. I should retest with the server power supply.
Regarding the RC ESC, I did not know that would defeat the PMBO spirit, the ESC is putting out pulses, is that true? (I currently am not, but I did try it last year and couldn't get it to work, well, I removed the mpu, and was interfacing with the fet driver and also using its fets, but never got the fets to turn on, doh!)
Air core of course is not nearly as efficient, but in my project, it seems mandatory to use air core, simply because I haven't cracked the code to using ferrite core issue, ball just sticks to the ferrite, haha, iron core is too lossy at my +25KHz frequencies. Well I just thought of using good Japanese silicon steel as its much better at higher freq, but that still does not stop my ball from just sticking to it. :)
I have tried so many things that either did not work or gave no gain in preformance. I should of spent 5 minutes at the end of lunch hour to take notes on that days experiments.
There is so many different directions a designer can take, it's fun to experiment.