Follow the note name. Add accidentals later.

Begin by acknowledging that this pitch is some type of C4. Then add accidentals to show that it's C♭♭4.

This is a little strange, because (as you say) C♭♭4 is lower than B3. But that's no more odd than saying any C♭♭ is lower than its nearest B.

And the same is of course true going the other way: B♯3, despite being enharmonic to C4, is nevertheless identified in the "3" register.

Follow the note name. Add the accidentals later.

Begin by acknowledging that this pitch is some type of C4. Then add the accidentals to show that it's C♭♭4.

This is a little strange, because (as you say) C♭♭4 is lower than B3. But that's really no more odd than saying any C♭♭ is lower than its nearest B.

And the same is of course true going the other way: B♯3, despite being enharmonic to C4, is nevertheless identified in the "3" register.