If you can get the actual thumb hole area and attach it to a similarly-colored background piece (or even a piece of another similar-looking fabric, loosely woven), then you might be able to overlay it with a piece of white/neutral-color tulle and carefully stitch all of it in place with clear thread.
Of course, you might want to try this a time or two with other-than-precious fabric, cut away a bit of a small hole and work at fraying the hole edges before putting it between a foundation piece and tulle.
The tulle and clear thread will practically disappear when done, but it will hold everything in place, especially if the piece ever gets carefully washed.
Just a thought. Keep us all posted on how this precious project ends up.