Now mind you, I AM run something "close" to your idea..
I have a HUGE 12v LiFePo4 (60Ah) battery that I use an inverter and a 36v charger to recharge my 36v battery while I'm running, but it's about 80% efficient.
(I got the 60Ah 12v for a THIRD of it's normal price - $125 instead of $375 AND I already had a 200w inverter).