Considering that the remedy to the emissions fail is so simple, this does appear that the Mot tester was being less than helpful. However, being fair minded he/she may not have any knowledge of A series carburation and it was a Friday ! You were certainly unlucky especially considering the multitude of things the antiquated Mini can fail on.
The engine is burning (slightly) too much fuel as the exhaust analysis shows. As 1984Mini25 says there is a nut at the bottom of the carb. which can ( just ) be turned between the fingers which when screwed into the carb, anti - clockwise when viewed from above, will weaken ( reduce the petrol) the mixture. This may result in the car becoming a pig to drive but so be it, it will get you through the test and you can adjust it back afterwards.
In the 50 odd years I have been on the receiving end of Mot's I have ALWAYS accompanied the car through the test. Most testers are more than reasonable and as long as you do not interrupt them and only respond when spoken to I have found them always helpful. I have had them volunteer to adjust the mixture to achieve a pass providing there are no other issues. Better luck next time.
Edited by rww, 19 May 2017 - 06:55 PM.