What is the best vinegar to use to make greengage chutney and why?

I would use cider or white wine vinegar. I find distilled vinegar too harsh and other specialty vinegars like balsamic or malt have a unique taste that may transfer to the chutney. Rice vinegar would be too subtle and red wine vinegar will add an unwanted color.

So, in general, I would use a white fruit based vinegar.

Greengage Chutney

I would lean towards the cider, malt can be to strong for some fruits and the chutney will be very vinegary in flavour, it also depend on the sweetener you use, regular sugar, brown sugar, muscavado or Indian Gur or Jaggery, plain white vinegar would be my choice if you were not to use the malt or cider, I make mine in the fall here in Canada or when our local peaches come out in August I may make a peach and yellow shiro plum one, greengage are a bit more tart, we do get a type of them here, my chef mate in Liverpool even does his own pickled walnuts, his gran has big tree in her yard in Essex.