You don’t have to change the „functionality is NOT guaranteed“ with „Guaranteed to blow up your house“

but there are some issues. In case I didn’t miss something, here are my findings:
*The maximum charge voltage, the battery sees, is about V+ Charger minus 0,7V (D1). How should IC1C switch, when V Battery will never get higher than V Charger ... except if the charger is disconnected?
* I’m not completely sure about your intentions but I guess that the Zener diodes won’t do what you want them to do. If you want to compare the battery voltage to a fixed 6,3V and 5,6V, you need to put D2 in the place of R2 and D3 in place of R5. As you’ve connected them, they would subtract the Zener voltage but only if current is flowing through them. Because the comparator’s inputs have practically infinite resistance, no current will flow and the non-inverting input will see the voltage of the voltage divider R1/R2 and R6/R5.
* The LM339 has open collector outputs. This means that it can sink current (pull the output low) while –In > +In, but whenever +In > -In, the output is floating and you need additional pull-up resistors to drive Q2 and T1 and to make sure Q1 shuts down.
* VOUT2 is connected to a FET and a transistor at the same time. The base-emitter-voltage of T1 is about 0,7V, the gate-source-voltage of Q2 should be several volts and therefore Q2 would never turn on in this setup. Either find a good combination of pull-up and T1 base resistor, or to make it simpler for you, use the fourth LM339 with inverting and non-inverting inputs of IC1B and IC1D in parallel and one output driving the transistor, the other the FET. Still don’t forget the appropriate pull-ups.
* The dead battery indicator seems to be an logic AND of VOUT1 and VOUT2 (LED is lit when both VOUT1 and VOUT2 are high). Why? If the battery has dropped below 5,6V, it has definitely already dropped below 6,3V. Connect R4 to VBattery and forget VOUT1.
* Maybe there is some use for VOut1. Whenever Q2 disconnects the load due to a dead battery, the battery may recover a bit and the voltage may rise above 5,6V. Q2 will reconnect again, the voltage drops and you have a nice oscillator. Just not sure if the device will love it.

Add a Flip Flop, which will be set if the battery voltage is above 6,3V and reset if it is below 5,6V. This way it won’t oscillate.
There might be one or another point but don't stop to experiment. The start is always hard.