Bmw legal issues

I got a 2003 BMW 325i a few months back from a dealership that used a 3rd party bank to finance from. That bank offered a 5 year or 100,000 mile warranty on the transmission. After about 3 months of having the car, I had transmission problems. The car got towed to a repair shop and an inspector from the bank went and investigated the car to see if the damages were covered under warranty. They denied fixing it because apparently the cooling system caused the trans to fail.. Now Im no mechanic, but is that even possible?? Doesnt sound right. So im thinking they just said that to get out of having to pay for it. I argued about it for a while, so finally they said they would be $1,000 as would the dealership. Leaving me a $1000 left over for ME to pay. Of course I didnt have that money to just fork over all of a sudden. NOW this is my real problem. Legally, can they do this?? At the time of signing, I was never made aware of these "terms and conditions" that they say I dont qualify for. And I quote "if an outside problem (cooling system) causes the tranny to fail, we wont fix it"... I never heard, seen, or signed anything agreeing to that. But they all of a sudden they have these terms when something does go wrong.. So I really need some advice here.. What should I do????