question

Is the shunt wired correctly?
Only the shunt is connected to the battery negative all other negative have to be connected to the shunt.

Can you show us the settings of the BMV?
Maybe some pictures of the wiring.

Yep, i think the BMV was not set up entirly. I changed a few things with the smart dongle connected to the BMV 702 ( see picture)

I will try to charge the battery until 14V and then manually set the current state-of-charge to 100%.

I think my battery is rated for a maximum of 14.6 V. But i guess it is not that good to go all the way to the maximum.

I think the wiring is fine, only the negative of the battery on the battery only side of the shunt, the rest of the negatives on Load and charger side of the shunt

Yeah, the peukert exponent was 1.25, the charged voltage 13.2 V, and maybe that fridge made it worst on top of that, with an incorrect SOC also at the begining.

here is the settings and the wiring too.

No point in manually setting SOC. It will reset to 100% automatically based on reaching the charged voltage and the current dropping to tail current for charge detection time.

Could it be that your discharge floor is 50% which I believe is only for agm/gel/lead acid where 50% is usable.

I'm pretty sure for lithium it is generally 20% discharge floor or 80% of stated capacity (manufacturer dependent). This could explain why it is not utilising the full capacity. I had the same problem when setting my system up and this solved my problem.