I believe that all Burgess pumps use the pistol grip pump action. I've never seen any other kind and I have been looking for a while and own one of the takedown shotguns. I have handled 2 others. There is rare (the Burgess takedown pump shotgun), "more rare", (the Burgess folding shotguns with the hinged barrel and magazine), and "really rare" (Burgess pump action rifles built on scaled down pistol grip pump actions). I have only ever seen pictures of one of the rifles, on 24 Hour Campfire.
My guess is that the only way to fix your Burgess is to have the gunsmith take it apart, figure out how it is supposed to work, and then determine what is not working right and fix it. Likely to cost some bucks. The lifter on my Burgess has been worked on at some time over the years, so that may be a place to start looking for your problem.