There is a possibility cutting the root will cause the tree to decline. If it's a black locust (Robinia pseudoacacia) it will be likely to produce nuisance suckers and bothersome thorns anyway, making it a possible candidate for replacement. But from you description maybe it is actually a honeylocust (Gleditsia triacanthos).

For successful retention of the tree ideally the paving would be reset with a new level that accommodates the root. It's impossible to say whether or not cutting that particular root off will spoil the tree.