It's not easy. You can do a few minor Legion missions until you're liked, without ever going to the strip. Get the safehouse key, loot it, then go to the strip and your status will reset when Crocker contacts you. Then side with the NCR. Be careful, as some Legion quests will make some NCR quests impossible to complete.
HAXXXX! that is how. due to the fact it is not giving to you until you become liked by Caesar. I tried killing him for it that doesnt work.
Caesar doesn't have the key, Lucius does, but it's not in his inventory. He spawns it when he gives it to you.