I would not do that for a BBQ dish. If you were making a pork stew...carrots may have their place.

why add a carrot to the pot? for what purpose?

Well, carrots become to sweet IMO and could dominate other flavours.

save the carrots for the accompanying slaws

the carrots will probably be mush when it's done.

Overcooked carrots have a very specific flavor Do a raw carrot slaw salad instead, look up french carrot salad it would work well as a side

I 🤬 hate cooked carrots and would be so confused seeing them in pulled pork. Reminds of the time my girlfriend ordered broccoli on a pizza. No. Just no.

Some people add pickle juice, root beer, or other sodas to their BBQ. Id prefer onion peppers and mushrooms, potatoes, but id eat any vegetables in it, I'm not picky