£60 extra for cassette and fitting sounds a lot? What groupset is it?
If you only change the chain, you may hasten it’s demise on a worn cassette but up to you. It’ll just mean another £20 potentially but you could the. source the parts cheaper online and ask a shop to fit (mine only charges another fiver if I get the parts)
I am not saying it is cheap/expensive because have never bought it, so don't have a reference to compare. I am concerned as that is not my regular bike shop and in some areas of London they try to scam you easily, that's why I was asking. I don't know the chain he is putting, but I saw the cassette was shimano of 9 speeds...nothing more that I know. I will probably keep an eye on it and will take it to my trusted regular bike shop. But thanks!