Ailish, unless there is a warranty, 12 months is a long time and taking an aggressive stance with the garage is probably not the best way to handle this. They could argue that they didn’t now about the accident or that the pipe has been damaged since they sold the car.
If you use the garage for services and repairs I would call in to see the service manager. Explain what has happened and ask if, in the interests of them retaining your future custom, they would be prepared to waive the labour costs for the repair. This means you will just have to pay for the pipe and air conditioning gas. This is not perfect, but will save you a lot of money and time-consuming argument.