You're misunderstanding the Doomsday argument. There is no cap, nor does humanity have to disappear overnight. In fact it makes no claims about how humans will go extinct.
What is the correct interpretation of the "doomsday" but the point at which our population growth becomes statistically likely to stop. If we have any other model for population growth, the ability to finger a likely date for maximum population tends towards zero. What am I misunderstanding?