Thanks all - perhaps I'm reading too much into the 'deep' breathing. . . .
What is considered an acceptable fluctuation from a stop depth due to breathing? 0.5m either side? Can you ultimately get to a point where you don't move at all?
It's common to see advice to breathe "deeply," but it's simply a matter of poor wording. Breathing shallowly during a dive is bad, as it can cause you to retain CO2, potentially resulting in headaches among other problems. (Whether it affects offgassing seems dubious--if I recall, that has been discussed on SB and debunked.) Advice to breathe "deeply" is intended as a way to emphasize "not shallowly." The best advice I received is what others here have said: take normal, full breaths, whether at a safety stop or any other point in a dive. It helps the body properly exchange gases and helps with depth control.
You should rise and fall almost imperceptibly with each inhalation and exhalation. Just a few inches each side--so little that if you're watching some floating particle you can keep your eye on it the whole time, watching it seemingly bob up and down. (That's actually a great way to practice.)
Holding a stop in shallow water is one of the hardest skills to master. I thought I had gotten good at it after all my years diving a tropical rig, and then when I started diving a steel twinset I realized I hadn't mastered it at all and spent several more years working on it on a regular basis. The more massive the tanks, the more it's important to keep your depth under control. Once you lose control and begin inadvertently ascending or descending, you have to fight the momentum of the tanks to arrest the ascent or descent.