I made a pair of overmitts many years ago. The original plan was to sew a piece of lycra edging strip across the inside of the back of the hand (i.e. inside the mitt) , but in the end, I simply sewed the ends of the strip into the side seams, and didn't bother with attaching the strip any further. It worked just fine.
For a pre-existing mitt, you'd have to open up the side seam, insert the edging strip, and then re-sew and re-seal.
An alternative method might be to try using iron-on seam seal tape to secure the strip to the mitt; use a piece of seam tape either side of the lycra edging strip, at both ends, and iron in place.
I can't guarantee how long the adhesive will last, or how well it will stick (DWR has an effect here). But you wouldn't have to attack the mitt with seam ripper or scissors...