Yes but it won't work.
- Yes, because you just make a selection of a thin slice of your gradient, and copy it to the clipboard, and then use the Clipboard brush (first in the list).
- No, because things are not going to be pretty in curves. To paint with a brush, Gimp just repeatedly hammers regularly spaced copies of the brush. which is kept untouched, while curves would require one side of the brush to shrink and one side to expand. If the curves are much wider than the brush size, you can be lucky if you give your brush a butterfly shape, but it doesn't work too well in the general case.