I have no comment on breath of fire, but wanted to say that sometimes my breath is affected not from the physical exertion but when I come to a pose I already think is challenging I exert more than necessary into the pose which changes my breathing. If I back out, stop thinking of it as a challenge (this is the most important part because if I cannot get to this place then going back into pose is pointles) and then slowly join back into the pose. If I go into the pose in a softer state of mind my body follows suit. Sometimes I do need to back out and that was my problem, it is much easier to see this coming from a place of self compassion for my body. But a lot of the time it is my mind that is pushing and not even my body because my body can go as far or even farther if I am in the right state of mind.
For instance, if you have a stressful day at a job where you sit all day. You can just sit and hold all of your body muscles tight. Your jaw, and arms, and core and legs. At the end of the day you will be very sore and exhausted, your breathing with suffer, you will probably get a headache, neckache etc. If you do this all the time you can get chronic problems. And all you did was sit!
This has been my most important lesson from yoga. Before I got it a teacher once told me to stop worshipping the body and start worshipping my mind. We do all sort of strange things to worship the body, push it to hard to show its strength, go easy on it to give it a break, and just give into the physical pleasure it can provide. Sometimes we even punish it in order to worship it. I am not sure how we came about doing this. I am not saying that you can not have a strong and pleasurable or relaxing and pleasurable practice. I am just that if you get it from the mind, it will be a more true practice to who you are and what you need and ussually the results are more what you want.