we did.. they said we have no proof of the car not starting

hello my husband I have recently purchased a 2004 bmw for £3500 29 days ago from a used car garage. About a week ago, the car would not start. We called the dealer and he came out to look at the car. He ran diagnostics and said nothing was wrong with it, that sometimes the sensor causes the car not to start. He had to put it in neutral a few times and started it after a few tries. It worked for a few days, and then yesterday the same thing happened. The car wouldn’t start so my husband tried doing the trick the dealer was doing but still nothing happened. We were stranded without a car yesterday and called them to let them know the car wasn’t working, and they said they would call us back in a few hours. They never ended up calling us back even after we sent texts to their cell and called the dealership. Today we got the car to start, so we took it to another mechanic to have them check it out and they told us they sold us a faulty car. The diagnostic test said: CD99 Telegram Monitoring (Torque Requirement Electronic Transmission Control (EGS), B5) CD9D Telegram Monitoring (Gearbox Data, BA) We then went to the original dealer to request a refund for the car and they refused. They said we don’t have the right to return it even if it’s within 30 days. He said they will fix it, and if it’s not fixable they can give us our money back. We told him we don’t want it fixed, he already “fixed” it once and it broke down a few days later. Is this a case worth fighting? New to buying cars and not 100% sure on my rights.