Oh, you're not going to like most of what I'm going to say. But it will be fun, entertaining, and educational!
Truck batteries and marine batteries are exactly the same, the only difference being the terminals. They are both "hybrid" starting/cycling batteries, meaning they have thicker lead plates than a starting battery, but thinner plates than a true deep cycle battery. They are roughly 100 amp hours batteries, measured at the 20 hour rate, which is at a 5 amp draw.
Because of something called the Peukert Effect, the rate of amp draw from a battery is very much like the fuel economy of a vehicle, where the faster you draw amps the worse your fuel mileage (available amps) will be. If you can draw amps out at a 5 amp rate, you'll have 100 amps available. If you draw amps at a 2 amp rate you'll have more than 100 available, and if you draw out 10 amps you'll have considerably fewer amps available.
Amps x volts = Watts.
Watts / volts = amps.
125 Watts / 12 volts = 10.4 amps
Adding 10% for the inverter loss (1.04) and you're at a 11.46 amp draw.
That's a VERY high amp draw for a 100 Ah battery. And because you don't want to draw down the batteries more then 50% to keep their lifespan as long as possible, you'll need at least a 200 Ah battery (or battery bank) to last you 10 hours at an 11 amp draw.
An "only 65w" fridge is still about a 5.5 amp draw from a 12v battery (its also what those electric coolers like the Igloo coolers draw). That's the maximum sustained draw that a 100 Ah hybrid battery is designed to do. And even that's not really designed to be a virtually 24/7 draw. It's really more for the occasional 5 amp draw of a trolling motor or running lights for short periods of time (an hour here, an hour there). But if that's all you EVER connect to it, you'll be OK, and you'll get probably 36 months or more or of the battery. Increase that amp draw to 10 and you're looking at 12-18 months max before the battery loses all of its capacity. Double it to 200 Ah and you're looking at 18-24 months, and triple it to 300 Ah and you're at 36 months or more.
Using batteries with true deep cycle construction (not truck, not marine) with thick lead plates designed specifically for sustained amp draws will last much longer, perform more efficiently and week be more economical. Look for floor scrubber batteries, aerial lift batteries, golf cart batteries. These are true deep cycle batteries. Never skimp on tires or batteries. You get what you pay for.
When you plug your hair dryer into the electrical socket at home, you can plug it in professionally, or unprofessionally, but it gets plugged into the socket the same way regardless. Installing aux batteries are the same thing. You can install them professionally or unprofessionally, but the connections are exactly the same regardless. In the cable between the starting battery and the aux battery you want a battery isolator (or a battery combiner, depending on the type of alternator you have) so that when the ignition is off the starting battery and aux battery are no connected together (they are isolated from each other). That way when you are parked and running inverter loads you aren't putting the same amp draw stresses on a starting battery that's not at all designed for inverter loads. You will also need to ground the aux battery to the vehicle chassis.
Does anyone know of some sort of battery that I can plug into the cigarette outlet to charge while the van runs?
Lithium ion batteries, like the one in your smartphone, can be charged from the cigarette lighter socket. They make vehicle charges for that very purpose. Those AA and AAA rechargeable battery chargers, if they have a 12v plug, can be charged via the 12v socket.
Hey, I told you it would be entertaining.
But no, a 12v dash socket really can't be used to recharge an aux battery bank. You need at least a AWG 4 gauge cable (or thicker is better) from the alternator to the aux battery. The wires from the fuse block to the 12v dash socket are probably 14-18 gauge. Trying to charge a 100 Ah battery using those wires will result in at the very least a blown fuse, and likely a fire under the dash. I realize it's not your vehicle, but still. That would be embarrassing. Not to mention expensive. And maybe dangerous.
A 1500w inverter is going to require AWG 6 cable from the inverter to the battery bank, and that's assuming 3 feet or less. The longer the cable the more the voltage drop. Cables from the alternator to the aux battery will need to be much thicker. Don't even entertain the thought of using 16 gauge wire for that.
