Scott,
Doug is right. All Ruger SA grip frames will interchange from the standpoint of the screw pattern from single six to Super BH and from Old Model to New Model.
Your swap is as easy as they come. There should be minimum of fitting if any because the birdshead is the same relative vintrage as your BH. So the ear height around the hammer of the grip frames are the same. The sides of the frame will be very close if you get a used one. But new grip frames need a lot of fitting to match the sides of the frame and then the whole grip frame needs polishing and bluing. If your new BH has a lock you'll need to use the standard hammer spring, strut and spring retainer that comes with the Birdshead grip. All your other grip parts can be used.
I prefer the Cimarron birdshead because of the hump behind the hammer which is authentic to the Colt Lightening & Thunderer. But they won't interchange with Rugers because of a different screw pattern and the hammer spring.