It's about as civilized as a social place in Natchez is going to get. You might call them to make absolutely sure they have a TV in the bar: 601.446.8500
Here's their website.
I'm an old fogey in a lot of respects. If I just needed somewhere to get a cold drink and watch the game on TV, I'd probably go to the bar in the Eola Hotel where the crowd would likely consist of me, the bar tender and two other people who don't know there's a game going on. The Eola is downtown at <span style="visibility: visible;" id="main"><span style="visibility: visible;" id="search">110 North Pearl Street.</span></span>
Fat Mama's Tamales has a fairly-new and much-expanded location across the street from their original spot now, and I think there may be TVs at their bar. This place is also downtown at <span style="visibility: visible;" id="main"><span style="visibility: visible;" id="search">303 South Canal Street.</span></span>
Any of these would probably do OK on your low-smoke request. I don't know of a sho-nuff sports bar that would qualify, but I'm only there a couple times a year.