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Does anyone have any experience of dealing with Universal Music Publishing Group when trying to get permission to arrange a piece of music they hold the copyright on?

I've dealt with publishers before but not UMPG, specifically. They will try to get as much money from you as they can. Plead poverty, amateur performance, small audience, etc. Anything you can think of. They'll likely bend.

The rules change if you're planning on recording the arrangement. Check online resources.