Elaboration may be difficult without an understanding of how you already understand the third tetrad of anapanasati, or cittanupassana generally. Maybe you could also say a few brief words about your shikantaza practice? In any event, some brief thoughts come to mind, a bit like a scatter plot...
You had said "stay focused on the breath". As far as I understand things, the third tetrad does not employ a narrow or laser-like focus on the breath. As the mind is found to be one way or another on the inhale, so it is noted. And on the exhale, so noted. This is experiencing it, per the first instruction of the third tetrad - and the breath is simply like a metronome, or a ticking clock. Not ignored, not made a focus - and thus, your problem is averted, I think.
The mind can then be gladdened through appropriate means & antidotes, calmed, and then released, but throughout the breath ticks away. It isn't even the case that I consider whether it's the same mind or different minds over time - this would just be papanca, I think, in the context. Instead, the point is that for the in-breath, just this is my worktable, my workzone, for bringing mindfulness to bear on the mind (or whichever category). Then, when it is the out-breath, just this is my worktable, my workzone.
This awareness of breathing, which is to say, the presence of breathing within whatever scope of awareness obtains, persists even through most jhana. But it's not the focus.