Hey Matthieu, thanks for using WindWorks. I had this exact problem when I was younger.
I suggest you never have this problem when using the voice talking or singing.
Over time, you will learn to only take the amount of air required to get through the phrases you need to play.
The BackSwing and Body’s Concert Hall exercises are for creating healthy breathing habits and optimising your capacity. That being said, you don’t need to take a full breath every time you play. As long as you are in the Positive Pressure Zone every time you go to play, then the actual quantity of air is purely dependant on the music itself.
Breathing fresh air on top of stale air is very uncomfortable and I know the feeling of having to stop, breath out then start again. To solve this, as I said, learn to take the appropriate amount of air in. This comes with experience and knowing how much air you need.
Use the WindBag and do the exercises to learn how to get a large amount of air into the lungs super fast.