Alec, I have done some work on the sim, added more of the circuit so I could study the timings etc and I find this circuit acts pretty much the same as the charger, just play with the battery voltage to see the different comparators trip, changing the voltage of the gate drive just changes the charge current as in the real circuit I have it giving about 3 amps similar to the low mode.
I see the top circuit trips and stops the charging at around 14 volts ish, the bottom circuit is strange and only trips if the battery voltage is about 2 volts or lower, yet as I said before if you completely disconnect the battery and put a capacitor between the leads it will charge it? I still can't really see a sensible purpose for the bottom circuit??
I feel the voltage on the battery is a key part of the circuit and the circuit did not start to behave like the real charger until I included the battery and output fet.
Please give me any observations, I checked all the values and found that C9 is 22uf not 220uf and as was said before R6 is 1k6