It definitely is related to battery status--starts fine on a fully-charged battery.
I have a modern AGM battery (205 CCA).
Sounds like my suspicion that it's the sprag clutch (and that it's kind of a pain to fix) is correct.
I just replaced the spacer behind the alternator rotor in my 1998 ST2, so I am pretty familiar with what's needed to get down to the starter sprag, including tightening the alternator nut properly at the end
. Once the alternator is off, it's not much more work to get to the sprag.
Thanks for the input.