It is the sixth Games in a row that GB have been in the third and fourth play-off. They will tackle Spain for bronze on Sunday (September 5) after they lost their semi-final to the United States.
GB were favourites for gold having won the World and European Championships. Manning did his bit with 16 points against Japan.
Peterborough & Nene Valley shot put ace Lydia Church finished eighth in the F12 final in the athletics stadium.
Church threw a best of 11.41 in her second attempt to finish more than two metres outside the medals. A repeat of her pesonal best thrown of 12.18 would have been good enough for fourth place.
Stilton shooter Matt Skelhon competes in the mixed 50m air rifle event on Sunday.