Two separate problems get tangled together here: keeping the fridge cold, and keeping the station full. Solve them in that order and the wiring gets simple.
Put the fridge on the station, not the vehicle
A 12V compressor fridge plugged into the vehicle's accessory socket is drawing from the starter battery, and most vehicles kill that socket when the ignition is off — which is exactly when the fridge still needs to run. Plug the fridge into the station instead. The station has a real battery management system, it will not leave you unable to start the car, and the fridge sees a stable supply overnight.
Run the fridge from a DC output rather than through the AC inverter where you can. Going 12V to 120V and back to 12V wastes energy for no benefit, and inverter standby adds a constant draw on top.
Charge the station properly
The cigarette socket is the weak link. It is usually fused low, and it is why people arrive at camp with a station barely fuller than when they left. A dedicated alternator charger fixes it. Our BLUETTI Charger 1 listing states up to 560W while the engine runs, which Bluetti rates as six times faster than a cigarette lighter charger and enough to fill a 1kWh station in about two and a half hours. It switches on and off automatically so it will not flatten the starter battery, and it protects against short circuit, reverse connection, under-voltage and over-temperature.
If you also carry a folding panel, the Charger 2 does both at once: our listing states up to 800W from the alternator and up to 600W from solar simultaneously, 1,200W total into the station, with protection against draining the starter battery. Both are listed by Bluetti as working with all BLUETTI stations and most other brands.
Sizing the station
For a fridge plus phones, lights and a laptop, a 1kWh-class unit is usually the sweet spot. Our Elite 100 mini listing quotes 1,004.8Wh with 700W of output — enough for a portable fridge, a router, a TV, fans and laptops but explicitly not a microwave — with a quoted 34.4 hours running a portable fridge, and it lists the Charger 1 as one of its recharge options. If you want headroom for an induction hob or tools, step up in capacity rather than expecting a small unit to cover it.
What to check on your vehicle
Find your alternator's rating and whether the vehicle uses a smart or variable-output alternator, because that determines where the charger is fed from and how it is triggered. On a hybrid, check the manufacturer's guidance before connecting anything to the 12V system — Bluetti's own FAQ for the Charger 1 covers hybrid and EV use.
Genuine BLUETTI, shipped direct from Bluetti's US warehouse and sold by RV Parts Giant in Upland, California.