Change of possession means the first option that you listed - B getting the rebound of the 1-and-1.
If Team B had gotten the rebound and the timer had buzzed as they were crossing half-court, you would go shoot the final free throw with no players on the lane, and then award the ball to Team B for a spot throw-in on the sideline.
If Team A had gotten the rebound and the error was noticed before a change of possession (and within the time limitations of 2.10, of course) then you would shoot the final free throw with the players along the lane and "play on" from that point.
Whatever you do, make sure that you bring both coaches together and explain (briefly) what you are about to do and why. A good way to start is "By rule..."
