Once David had his disguise in place, he could easily make his way into the ranks of the Covenant, and sabotage things exactly as he saw fit.
Of course, there are some details that have us feeling that a completely separate solution is in play.
The obvious solution is that Michael Fassbender’s David won the fight he had with his double, Walter, in the cave-like dwelling that a good portion of Alien: Covenant’s action takes place in.
Technically, this shouldn’t be that much of a stretch, as an android could make almost all the changes needed in order to pull off such a deception.