Lithium batteries are dangerous (explosion or fire) if discharged to less than 3V then normally charged. Your idea has nothing to detect a low voltage then disconnect the load. I do not know if your charger will detect a low voltage then attempt to charge safely.I also have the option of supplying everything with 7.4v using two 18650 li ion rechargeable batteries.... would 7.4v to the opamp work as long as I divide the voltage to 3.7v+ into pin3? That could be the solution to this altogether.
Their voltage is 4.2V when fully charged so two cells produce 8.4V that might zap your DVR and their low voltage is 6V that is too low for many 5V regulators But a "low dropout" 5V regulator will work.