below link give idea of what and where it is on top of gearbox yep diy fairly easy
Year: 2004 - 2006 - Facelift Model fitted with a Getrag 5 speed gearbox. Make and Model: Mini One / Cooper. Speedy-Spares are an independent breakers yard who have been involved in the motor trade since 1960.
or if the pin gone below repair kit,
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not saying it s this just that if a garage has looked at it and is saying its something then normally it is what they tell you, that said all old cars are a can of worms never really know what you have until open them up, and any good tech might give you a number of outcomes based on other jobs etc, also based on stripping it and seeing what you really have, seen so may people do the you said it would be this or that when most gives a number of outcomes based on what is found, as no one has a crystal ball or can see in to the future etc,
if no one has looked at it its all guess work and thats the mother of all f ups