When a tennis player grunts as they hit a ball, this grunt helps with the effort. In the same way, we can use mukha bhastrika to help us with a few poses. In Surya Namaskara specifically. As you go to the floor as in astanga asana/8 points pose/knees chest chin to floor; or come down into chaturanga asana/low plank, whether from standing or plank; or just lower yourself, even gently, to lying flat on the floor; go down with one mukha bhastrika. And then as you go up into mountain/downdog, do another mukha bhastrika. This is particularly effective as a method to build up strength. Again - no overkill, please!!
An alternative version is to omit mukha bhastrika with Surya Namaskara as we described above, & instead hold mountain/downdog as in Surya Namaskra A, with or without plank pose included, & in downdog, do 1 - 5 rounnds of mukha bhastrika, in this way: inhale nose; exhale mouth in a bird's beak: shoo! shoo! shoo! 3 - 6 "shoos" per one inhale. This is wonderful for increasing daily alertness.
There is also a mouth bhastrika whereby one does bhastrika as normal, pumping the abdomen, but with a twist: do both inhale & exhale through the mouth. Do 20 or more. Powerful for alertness & cooling inflammation, especially in the gut. You can do bhastrika through the nose first, then do the same amount with the mouth, an equal number of times. Do this at the end of poses, in your pranayama section.
And another alternative, channelled by myself: inhale/nose; exhale/blow out through the mouth. This is not the mukha bhastrika that I have been describing, it's just a strong "blow", pulling in the abdomen at the same time. Do this fast, about 20 times, then follow with 20 fast kapalabhatis. Very nice. Do as your cleansing breaths in your pranayama section, without adding extra types of bhastrika.
Another type of bhastrika that I have successfully used in therapy, & it has been so strong in waking up the brain, is a hissing breath. For people who cannot do poses, they can sit, inhale/drop head; exhale/lift up head as you hiss the breath out loudly, look up at the end. For the rest of us: do this in cobra! As you raise head/shoulders/chest, hiss the breath out loudly. Inhale back to the floor. Do 3 -5 times. Not in Surya Namaskara, but as a pose on it's own. Be sure to look up as you go up. This also lifts the spirits, so is most useful, & is great for when we are feeling "down".
So, we now have several different ways to use bhastrika. Overkill is not advised. Make sure that you don't add too much to the poses. Always remember: less is more! & we want people to have a good experience, not a hyped-up one. Avoid bhastrikas in the evening, for they will keep you awake.
Do one thing & teach it for a while. Try to be consistent & not chop & change. Your students will love it. If there are any queries, please post in the comments section.