Brown sugar and maple syrup are two common tasting notes of bourbon, so the butter will only boost those natural flavors. It'll also add a smooth mouthfeel and some dairy richness.
Your best bet for making a warming fall and winter bourbon cocktail using TJ's sugar and syrup butter is to infuse the butter into the bourbon through the fat washing method.
Start by browning the butter to give it an even more complex toasty, caramelized nuttiness. Then, add it to some bourbon and refrigerate it all until the butter solidifies.
Skim off and toss the solids, and strain any extra fat that solidifies and separates. You can then use the bourbon, which will be left with buttery, fall-flavored notes, as usual.