Is it possible to not get the room at Lucky 38, but still complete and win the game? Is this faction-dependent? (I am specifically interested in an "independent New Vegas" solution, but am open to any suggestions. I'm finally on an NCR-centric approach, with Dead Money, but my next game will be independent with me killing everyone possible, and I'm more concerned about it then than now.)
Getting the room at the Lucky 38 in no way affects how you can end the game, the yes man quest path (which is the independent ending) actually requires you to have access to the Lucky 38 to complete it. Additionally, you can't lose the room even if you kill Mr House or do one of the other endings.
You could potentially complete the Legion questline. You would need to go straight to the Topps casino, and start Render Unto Caesar. However, I've never completed the Legion quests, so I'm not positive if there's a part in the Legion in which you're given the suite, but siding with the Legion, without stopping at the Lucky 38 would, at the very least, give you the longest amount of time one could go without that room.