Yes. Try the following:
Make the recipe as written but do not make it thick - shoot for the thickness of a thin syrup (so use the cornstarch slurry only if necessary, and only use a little of it at a time till the consistency is achieved). Since you like it, add a little teriyaki sauce - say 2-3 tablespoons - to the pot with the other ingredients. Adjust sweetness and salt as noted.
Purée the sauce in a blender till smooth when it's done. Reserve.
Rest the pork when it's done, as usual, foiled. When ready to pull the pork unwrap it, and if there are any juices in the foil add a tablespoon or two of them to the reserved sauce.
Reheat the sauce before use.
As you pull the pork, transfer it to a bowl or roasting pan and add 2-3 tablespoons of the sauce to every large handful of pulled pork, mixing/stirring it in as you go. Taste for salt and moistness when finished, adding a little more sauce if you'd like, and a little more salt - or a little Lawry's if you prefer - to taste. Serve.