Car Insurance

[Big_Lebowski]

Hello all,
I wanted to ask an advice to this forum.

I bought my second Kia car purposely for the very convincing 7 years long warranty. I don't like to change car often and I keep my cars normally for more than 10 years.

Now, I am at the 4th year and the air conditioning is not working anymore. The condenser (the "radiator" containing the coolant) leaks. The vendor claims that there is a hole due to a stone that hit it (and therefore that there is no warranty coverage). I have seen the radiator and I don't see any hole, but a laminated crack where there is also a lot of dirt and general signs of corrosion. That thing can be many things, but surely not an impact damage.
The 1300€ bill is uneasy to swallow and makes me reluctant to let go, but how to approach it? can I avoid to pay and start a litigation? I suspect they would never give me my car back if I don't pay, but I don't know.
Can I pay and start a litigation after? is there a way to contact Kia and tell that I want a higher inspection and that the car repair is not fair in his assessment? and how?

Any of you had experiences in these matters? Thank you for any help.

[Fraufruit]

I put your whole post into AI and got a very long answer that starts with paying under reserevation (unter Vorbehalt) to get your car back. Then there are a lot more steps. Makes sense and look legit. Instead of putting the whole wall of text here, please try what I did.