Not knowing what type of car it is, how many miles were on the car, or even whether it's a manual or automatic (assuming automatic), this is all I have for you ...
To easy your mind a little, any single 300km stint is not going to cause a transmission to go bad. The problem is, the transmission chose your 300km stint to break. This, in essence, leaves you holding the bag, as they say. It happened on your watch, so they are wanting to make you pay for it.
Something to think about here is who is saying you're liable? Is it the owner of the rental car and nobody else? This would sound suspect to me. There is no way he could prove you were negligent in your driving, which caused the issue. He is assuming you are at fault because it happened while you were renting the vehicle. The easiest and cheapest thing to do is to place the blame completely on your shoulders. He's wanting you to bight on this and you to pay all of the repair costs. It's the easiest way for him to go about it and the easiest way to recoup the funds.
There are some approaches you could use to mitigate some, if not all, of the cost, though.
First off, reread your rental agreement. It may be plainly obvious in the agreement who is responsible for vehicle breakdown. This is quite different than damage to the vehicle. Damage is intentional. The owner needs to prove you are the cause of the breakdown for it to become intentional damage.
Second off, as @JuannStrauss has stated ... GET A LAWYER (barrister, attorney, whatever you want to call it where you are at.) Sometimes just getting a lawyer to drop the company a letter stating they are on the job is enough to get a company to cease and desist.
Depending on the make/model/year/mileage of the vehicle, it could very possibly still be under warranty from the vehicle manufacturer. If so, the manufacturer could replace it under that warranty, letting you off the hook.
Get a second opinion. Have your mechanic take a look at the transmission and see what's going on. Things like burnt transmission fluid, lack of transmission fluid, leaks, etc., would tell a mechanic that this has been an ongoing problem ... something the owner should have been taking care of in the first place. If your mechanic can tell you that, I would sue the owner of the car for leaving you stranded and for lost time due to negligence. (This may be an approach you would want to take anyway ... ask the lawyer. I t would depend on the rental agreement.)
If all else fails, get a bat and break the dudes knee caps. While this will ultimately end you in jail, it will most likely make you feel better and the guy will think twice about doing something like this again.
Obviously, the last suggesting is a joke ... please don't land yourself in jail. All-in-all, I doubt you are responsible either. It is my approximation you are not to blame. I think there are ways around this quite easily ... but first of all, get the lawyer. He'll help immensely.