If they allow the price to rise, bar purchasers move to alternatives. (Whether independently or in conjunction with a shift at the distributor level.)
Currently, Beam can and does (per the article) use Makers Mark as leverage to sell their other products. I would imagine including before the point of the bar's purchase decision. Liquor distribution is big business in the U.S.; in many places, it is an oligopoly if not monopoly. (Illinois, for example, currently has issues with this. And I realize that this circumstance might argue for any potential change taking place at the distributor level rather than the level of the individual bar, depending upon whether and how much diversity the distributors offer.)
If you want Makers Mark, it comes as a package deal along with X quantity of this other stuff. Being that the other stuff is targeted at less discerning market segments (at the least, based upon price), a pretty good argument can be made that "whatever you buy for these segments, you'll be able to move it" (especially with a little effective marketing).
One of the first words that popped into my own mind was "bundling". Whether the article has the situation right, this is how I understood what it is saying. (With "bundling" being my word and analogy, whether or not it would be the proper term and analogy.)
I'll reiterate though, this is not an uncommon position for a bourbon brand to be in. Bulleit is a Diageo brand; one can only imagine the crap Diageo might want to bundle. Even Sazerac has a bunch of crappy mixers and vodkas in their portfolio.
I agree with you about price vs. demand; I think if you read between the lines what Beam already said, they're more or less acknowledging this (in that you can always solve a demand problem by increasing the price, and they're choosing not to do that).