Only thing to watch with shredding is the prickles are still there! No good for composting, I'd be careful with animals bedding too, even though they eat prickly things they're bound to get a thorn in the wrong place.
Burning - good plan, a bonfire seems like a waste, why not trim it and dry it out for the wood burner? Shredded wood is hard to get dry, or even stop from rotting if it's left in a pile.
PS Have equally ridiculous neighbours, but people still burn coal round here so can't say much to us