Joshua, Carter wanted to swap two pieces, not flip one.
I don't have a pillow, but I don't think this is possible from the pictures I have seen. The middle layer pieces are all in positions fully determined by the colours of the centres. The pieces on the top and bottom faces may be swapped, but this is an even permutation which will not help in swapping a single pair.
The only other possibility I can see is that 4 of the centre pieces have the same colour, in which case you can do a slice move in that plane (an odd permutation on pieces since centres are indistinguishable) and solve it again.
Jaap's Puzzle Page:http://www.org2.com/jaap/puzzles