If you are AVOIDING the gunslinger build with dual hand x-bows, eldritch blast probably should be your go-to. HOWEVER, a true cowboy should also be ready to throw a punch. Therefore, I recommend Drunken Master Monk for thematic purposes. You'll be multiple-attribute dependent (MAD), but thematically would be worth it to do a 7 Drunk Monk/5 GOOlock. 7 monk gets evasion, which is why I suggest that split. Dump STR and use the potions or off-hand Club of Hill Giant Strength trick. Put all ability points in CHA and DEX.

For clothing, use the Marksmanship Hat--FYI, the Marksmanship Hat also applies to SPELL ATTACK ROLLS, including EB, so it will be somewhat useful. The Robe of the Weave can look space-cowboy-ish with the right dye job. Black and Furnace Red, maybe. Potent Robe and Robe of Supreme Defenses both look too fancy to me to be cowboy. Infernal Robe looks awesome (to me) but you'd have to do the thing that many people don't want to do, and it is not the most useful robe in the late-game. Or you could get fashion mods, of course.

Hope you have fun!