What you have described in your first paragraph is exactly what Heaven Hill is doing with Rittenhouse but they are interpreting "produced" to mean "distilled," are using the word "distilled" on their label, and are identifying DSP 354 -- the distillery -- as the point of production. Looking at how someone actually does something is often a pretty good way of getting a feel for what the regulators are requiring.
The thing is, there are so few BIBs that it seldom comes up.
We have no examples of the alternate interpretation you describe. Regulations are always subject to different interpretations. The word "vegetable" can be interpreted as describing catsup (famous Reagan-era ruling). But all the evidence we have says we're supposed to expect a bottled-in-bond label to tell us the DSP of the plant where the whiskey was distilled, not where it was entered into barrels.