Jasper I think the difficulty is that your dads finish, like the one using muratic acid and steel wool, or other finishes of that type tend to "ride on the surface" of the wood. Yes they penetrate somewhat, but if you were to cut the handle in half, the center wouldnt look the same as the outside (would be my guess)
The stabilizing process goes all the way thru the wood, and deposits a resin type material (MMA), which fills the cells and hardens the wood when it cures.
You can get your birdseye stabilized, but its the color that will be difficult to match IMO. If you wanted to go with a color choice from their menu, I don't think it would be a problem.
Larry