You are going to have to play the numbers. Unfortunately you are not in a position where you can say " I want to work here, here or there". when the regionals start to hiring again, apply at all of them. You will have to make up for that DUI with time, if they require 100ME, you might have to have 300-400ME to get them to call. I'm sure you will have to be more competitive than the competition to get that interview.
Once at the interview, have a good explanation, and most importantly, accept responsibility. " I made a mistake, I'm very sorry, this is what I learned". As far as the background check, not sure if everything shows up or just what you were convicted of...... By the time you start applying at the majors/box haulers/fractionals, you'll have 5-6 years of regional flying under your belt, you're DUI should not be as big a problem as it is right now.