Hate to have to say this but think you will have to cover it....
points to note are...
The only warranty that has to be offered is that the car is of good road worthy condition when sold ( unless advised otherwise)
If the belt was new then they can argue that it is a fault of the belt and not their fault as could not have been expected to know.
If they say only one month warranty then thats all you can get unless you can prove a existing fault or one that they should have been aware of. The key here will be the condition of the belt..... if it is badly worn then you could argue that they should have noticed this, however if it is in good condition, but just snapped without warning then its another matter.
Small claims court is a option but it will be for you to prove that they are at fault.... been there, very difficult! Get someone who really knows the law to look at the paperwork you have and take it from there..... Of course you could always try the 'good will' route and maybe offer to 50/50 it. Still not cheap, but a option.
Good luck