Using the 12v ciggie lighter, to run a 240v AC inverter, to run a 12v DC battery charger, to charge a 12v battery is going round the LONG way lol.
If you really want to charge a second battery, just use jumper cables. if the battery is hard mounted in the car, buy a dual battery kit, or even go the cheap solenoid option.
Using the cig lighter, inverter, battery charger option you are losing efficiency in each step.
EG... the battery charger will charge at 2 amps (lets say) thats 24w.
The 240v side will use a touch more than that (as nothing is 100% efficient) so lets say 30w.
To pull 30w out of your inverter, it will be using almost double that on the 12v side, or more as inverters are at best around 50% efficient.
so thats 60w. add to that some voltage drop losses etc. and you will find your using 5-8amps from your cigarette lighter to charge a 12v battery at 2 amps.....
Jumper leads will be your best option (just for emergencies as you stated.
As for if the battery is hard mounted in the car as a dual battery type setup.....dont skimp, do it right the first time, and you wont have problems, failures, a burning wreck where your car once was etc.
You can set up a 12v-12v charging system pretty cheap (under $100) that is safe, although not as flexible and failsafe as the big name systems.
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