Strictly speaking, the accidental ONLY applies to the corresponding pitch in the same octave, therefore you are a little incorrect when you say:
I know that if there is an A# in the treble clef staff (where it's normally natural), it changes all of the A's in the rest of that measure to A#.
I think it's dreadfully unclear if there is no cautionary accidental for the same note in a different octave, and the composer/arranger does not intend that note to be affected by the accidental in another octave, but any cautionary is for clarity only, it is not required according to "the rules":
“In modern usage a sign is valid for the note that it precedes (but not for the same note in octaves above or below) throughout the rest of the bar, unless expressly contradicted by another sign.”
(The Oxford Companion to Music, my emphasis)
Furthermore and to answer your question: accidentals do not carry through to different staves in a grand staff, so the A's of your example would remain natural.