This is something I didn't talk about specifically in my guide to proper breathing technique that definately bears some discussion. I talk alot about moving from relaxation to exertion, but in the case of a static hold excersize you are in constant exertion. So what now?
This falls in the category of breathing under sub-optimal conditions. A corrallary might be an actor who has to sing while playing a hunchbacked character. He has to find as much support as possible while maintaining a contorted body position. This type of work is very specific to the particular body position at hand. It often time takes experienced coaching to get the maximal release, but here are some tips:
-Be mindful of your muscles and try to isolate the ones necessary for the excersize. Often times we will subconsiously tense muscles that are not neceasry for a particular movement. This is unecessary stress that usually hinders breathing.
-If you are in an awkward body position, usually we can gain substantial freedom in the breath with a very slight modification of form. Your point of reference is what I call center in my article (or the neutral position).
-DO NOT stop your breathe. If you aren't breathing, it isn't natural. Even swimmers use a slow controlled release of breathe while under water (changing the rhythm of breath instead of stopping it). Your breathe should always be in motion. If you stop the breath, certain muscles groups have to freeze to achieve this. This causes tension, lack of balance, faster fatigue, etc.
Now specifically for your handstand:
Try doing a handstand with a wall assist. I want you to keep your head between your arms and look forward (same as if you were standing upright. Keep your legs as straight as possible (point your toes if you have to). BREATHE. Now try to isolate the muscles needed to keep you upright, relax everything else as much as possible and continue breathing. Really strive for awarenesss of the tension in your muscles. Then move away from the wall. Make sure to keep breathing throughout.
This form is really great training for a static handstand. It will be more difficult at first, but it will pay off in balance and control. If you start to walk on your hands, you are gonna start to arch your back at least a little and this will change the muscle tension. This will add difficulty in breathing but the same rules apply, find an awareness, modify the position, and breathe.
I hope this helps. Cheers!