Do not hold your breath.  If you hold your breath, most people fill their lungs with air and constrict their throat, which causes tension.  It also compresses your heart and your heart increases its rate to make up for the loss of volume.

Instead, stop or pause your breathing.  Take a breath and let it completely out as you relax into position.  Do not constrict your throat: let your airway be open.  You should be able to manage this for 15 to 20 seconds, enough to get off a shot.

This assumes that you are able to rest your elbow on your hip.  If your arm is supported against your side you have other considerations.

Problem with BPCS is you can shoot good scores and place well in matches without the offhand leg, then hit a plateau due to the chickens. 

I always Shot offhand, not very well, some years ago decided to improve. Way I went about it was put a bolt action iron sight 22 spotting scope & ammo in the trunk of my car. Worked out a deal with a local commercial indoor pistol range to let me shoot last bay away from the crazies. Normal routine was 4 x 5 Shot Targets every shot strike called and observed, then stopped, all sessions very short, several times a week. targets saved scores recorded.

Progressed well, then moved the routine over to a good quality 10 M Airgun in my basement workshop. Sometimes back to the ,22 & pistol range but found the Airgun just as useful and more accessible. Improved fairly fast then hit another plateau.

Tell you later how I broke through that one. 

Boats.

One more before I have to shut down. Problem with silhouette (and I was silhouette match director several types our club) is a miss is acceptable, most people accept they can’t hit them all and throw shots away.

Very good competitor who’s name any BPCS shooter knows told me “ you shoot ok, scores will improve if you can stop making mistakes”. This falls right into line with Harry Popes “ it’s not good shots that make the score, it’s the absence of bad shots”

Issue with BPCS chickens is hard to spot misses, makes it difficult to see what’s going on which feeds throwing away shots. For silhouettes I use paper targets exclusively for practice.  Spray painted on cardboard or scrap plywood exact size from stencils. That way you can call the shot and spot the strike.  Use a cork board with the critters painted on one color push pin call another with the actual strike. Schuetzen, what little bit I shoot, same routine paper target at the bench every shot called spotted and recorded with push pins.  

What you want to do is train your mind to pull the trigger at the right time. Only way to do that is think about what you saw and confirming with what actually happened. 

Good book, perhaps the best on competive shooting is David Tubbs “Highpower Rifle”he goes into preparation and match strategy, avoiding errors. Tubb is probably the best shooter ever and has a clear way of explaining. Avoid his “rifle shooter” which is more equipment than technique. 

Boats

The Russian book is probably the best available on target shooting of any kind. Position illustrations are invaluable. It’s often available on Amazon but expensive. I am mixed on what’s best Russian or Tubbs. Best get them both.

Boats 

Mine came from NRA, less than 10 dollars I bet. They are trending on Amazon now 50-60 dollars and well worth it. 
Tubbs book 25-30 and another essential .

Boats.