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Since the leave now/stay a little longer choice is important for choosing the route how do you still get on a route that requires the option that now starts the DLC? 

At the welcome party, get a drink and talk to Slayton. He will offer to let you leave early  (for lack of a better description), which will bring up a choice marked "Begin Wyatt DLC" or something like that. If you don't select that choice, the game continues as normal.