Cut a piece to width and gently sand it to a perfect fit. Then use a hand plane to trim it to shape. Then use your jig to recut it. Try to find some scrap of the same material as the body. With very careful fitting you shouldn't see the thing at all. Be sure to scrape the glue squeeze out away so the finish isn't affected.
BTW, there are small luthier planes that will trim the piece very nicely. Here's a pix. These are very small, but you can find larger. They are quite inexpensive.
The more I do, the less I accomplish.