I figured that i'd post a guide on how to get the absolutely awesome looking Blackjack duster in Lonesome Road for anyone still wondering, as I was able to do it. So here goes.

The Basics - How Ulysses' Dialogue is Decided

Ulysses' dialogue (and the duster you receive at the end) is based solely on your reputation with these 3 factions: NCR, Caesar's Legion, and The Strip, leaning the dialogue towards NCR, CL, or Mr. House, respectively. You get the Independent dialogue and duster when your reputation with all three of those factions is neutral or below. Your reputation with FotA and BoS don't matter. The main questline you are following does not matter.

Setting reputation backwards
Now, you might be in the same situation I was - You are Idolized by NCR, yet you want the Independent duster, and don't want to intentionally, permanently lower your rep with them. There is a simple solution - disguises! Your NCR and CL reputation effectively doesn't matter if you have a disguise. (I used NCR disguise, but I imagine CL would work as well). Simply put it on, and your rep with both factions is now neutral!

The Strip Reputation
Your reputation with the Strip is the only one you have to worry about, as an NCR disguise doesn't change your rep with the Strip. (CL disguise might, i'll research that) Unless there is a disguise I don't know of to reduce Strip reputation, (maybe BoS) you only need to worry about that reputation.There are only 4 quests that grant Strip reputation: Talent Pool (given after you talk to Tommi about an individual act), Classic Inspiration, Beyond the Beef, and The Moon Comes Over the Tower. The last one seems to be given when Emily force-starts dialogue with you, but that is not, on it's own, enough to put it beyond neutral.

When to Wear the Disguises
Now, you can be like me and wear the disguise throughout the whole DLC to be safe, but you don't have to. Ulysses initiates dialogue with the player at the following times:

After leaving the Hopeville (first) silo

When exiting the collapsed overpass tunnel onto the high road

Exiting the Sunstone Tower roof

Exiting the Cave of The Abaddon (might be the Boxwood hotel roof. In any event, it's when he takes ED-E from you, though it doesn't appear to be faction related)

And, of course, in his temple, at the end.

If you simply put on the disguise right before you hit those triggers, you should be good to go.

I hope this guide has been of use to anyone trying to get the Independent duster. I think it looks sweet and I can't help but wear it all the time. Thanks for reading (or not)