Sounds like tap dancing. I realize that laws are probably different there, but here, if your car was damaged by a malfunctioning parking garage gate crashing down on your hood, they'd handle the claim. You insurance company would fix your car, then pursue your claim against the parking garage operator. There's no difference between that than and the charging station damaging your car. Unfortunately, it may take a lawyer to make anyone take responsibility, and if the repairs only amount to a couple of thousand dollars, it's not worth it.
What I would do, if you absolutely can't get your insurance provider to handle the claim, is push the charging company hard to cover the repairs since they are the responsible party. Pay a mechanic look at the car and state that the charging station caused the problem. Send certified letters with return receipts making your claim. Document every interaction you have with Tesla, the charging company, etc... If and when that fails, take whatever info you've gathered, then take the whole thing to small claims.
I don't know if they have the equivalent of our small claims court in France, but I expect you might. The way it works is that the court takes cases under a certain value limit. You argue your case yourself before a judge. Neither side is supposed to be represented by council, and there's no jury. Generally, both sides just get up, tell their side of the story, presenting whatever evidence they have, and the judge decides the case on the spot.
There will be rules about filing the case, and notifying the defendant, which you want to follow precisely. If you do a good job, you'll probably win.