What about
Spoiler Benny. When you confront him, he knows who you are instantly.
That kind of rules out major facial reconstruction. Maybe just minor surgery around the place you were shot. Also, there are a couple dialoge options that can imply the character has been to certain places, so amnesia is also out.
Amnesia is not all black and white. You may lose some or most of your memories due to head trauma, and you may or may not recover some or all of your memories. Post-traumatic Amnesia is the most solid explanation in this case, because the Courier was shot in the head.
"Post-traumatic amnesia is generally due to a head injury (e.g., a fall, a knock on the head). Traumatic amnesia is often transient, but may be permanent of either anterograde, retrograde, or mixed type. The extent of the period covered by the amnesia is related to the degree of injury and may give an indication of the prognosis for recovery of other functions. Mild trauma, such as a car accident that results in no more than mild whiplash, might cause the occupant of a car to have no memory of the moments just before the accident due to a brief interruption in the short/long-term memory transfer mechanism. The sufferer may also lose knowledge of who people are, they may remember events, but will not remember faces of them."
- Wikipedia