Not all BBQ sauces contain tomatoes. Ask our frinds from the Carolinas- their sauce is mustard based. If you Google mustard bbq sauce recipes
or Carolina style bbq sauce recipes
you'll find many to pick from ... here is one
Another option would be an Asian BBQ sauce. Again, a Google search for asian bbq sauce recipes
turns up several options ... here is one
without tomatoes. You can probably find a small bottle in the ethnic foods section of your grocery store if you don't want to make your own.
The asian bbq sauce would probably be my first choice for this recipe, followed by hoisin sauce, mayo, yogurt, and then mustard bbq sauce - in that order, and I'm still questioning if the mustard sauce would really fit in with this recipe.
Since the bbq sauce is supplying a good percentage of moisture I would substitute something ...even if you just added 3 Tablespoons of water to the egg before you beat it and add to the mixture.