I would approach this from a different angle. The ingredients in Barbie's are essentially the same as in Bush's writ large. In other words, though the recipe starts with Bush's, the additional ingredients are prety much the same as what is already in the can.
So, that said, I would not try to recreate Bush's at first, but would instead make the beans from scratch, plain, using navy beans, though Great Northerns or cannellinis would work too. Cook the beans till just tender (in a pot or pressure cooker) then use them in place of the Bush's. To me, this approach would likely result in a better finish because the beans will be less sweet (Bush's are already sweet and the additional sweeteners in Barbie's just add to that). If desired, one could up the sweeteners and other flavors, as necessary, near the time when the beans are finishing.
There are a couple procedural details I would change. There is little in the way of flavor development (in terms of depth) that occurs if one simply combines everything in the pot, as directed in the recipe, and cooks thus. This might work okay using Bush's because they've covered some of this in the process they use to make their beans. However, if using cooked dried beans in place of the Bush's, one needs to go for depth early on.
By doing the following, I think it is likely that one could make the beans pretty much as written, but using cooked dried beans instead, and also avoid the need to make ingredient additions near the end.
Cook the dried beans till just tender adding salt near the end of the cooking process to that they are slightly seasoned. Meanwhile, increase the dried mustard to 4 tsp and mix it with an equal amount of water. Allow to sit 10 min then whisk in 1 tblsp apple cider vinegar; reserve.
In the pot will you will later use for the beans (after the dried ones have cooked), add the bacon and render its fat over med-low heat, stirring often. Using a slotted spoon, remove the bacon from the pan and reserve. Finely chop the onions and add them to the bacon fat in the pot. Increase the temp to medium and cook, stirring occasionally, till the onions are soft and just beginning to caramelize in spots, about 20 min or so.
Add the diced peppers to the pot and increase the heat a little. (Consider adding 1/2 a green bell or Anaheim pepper--minced--as well.) Cook 2-3 min, stirring constantly.
Run 3 cloves of garlic through a press. Increase the heat to high, wait a min, then add the garlic. Stir constantly and cook till the garlic is very fragrant, about 30 secs. Immediately remove the pot from the heat and stir in the reserved mustard-vinegar mixture and the reserved bacon. Stir in the BBQ sauce, black pepper and Chimayo.
When the dried beans are just tender, measure out the correct mount and add them to the pot along with some of their water. Add the pulled pork, the syrup, molasses and brown sugar. Add the hot sauces, stir, then adjust salt, if needed. Adjust liquid by adding more of the bean cooking water if necessary.
Cook/smoke the beans as directed in the recipe, stirring periodically, and adding water if necessary to maintain/achieve the desired consistency.
If necessary (and I think it might well not be, or perhaps just on a small scale) adjust sweetener(s) and spicing.