Use of Air button
John, I went back and reviewed the reverse fingering exercise of Shoes and Stockings and I can see how the air issue could be a problem for you. Specifically the out movement in the ending measure 4 and out movement of the beginning of measure 1 (and measure 8 and 5) could create a feeling of instability in the centering of your instrument Here are some suggestions.
Try to practice the exercise in two ways: with and without the air button. First try to practice the piece without using the air button by planning out bellow excursion as previously discussed. This method requires that you carefully prepare the hand before executing the phrase as also discussed. This is the most important point of the exercise. If it is still a problem for you, simply play the last G of measure 4 and 8 (the last beat) in the closing position. You can use either the row II G (L10) or row III G (L13).
The second method is to use the air button to reestablish the center of your instrument. I would use the air button on the last beat of the second measure (the C) and on the last beat of the 6th measure (the D). You could either use the air button simultaneously while playing the note or shortening the note with a small rest. This is easier with the second measure since the C is a quarter note while the D in the sixth measure is an eighth note.
Of course the reverse direction version is only an exercise to learn to play the same notes fluidly in both directions. As I pointed out in the conclusion on page 38, once you have internalized both directions carefully, you will be mixing the two versions, which will ameliorate any bellow length issues.