Wouldn't Auto-Relist do both, i.e. "Don't Auto-Relist based on items sold" + "Relist with no changes" + "Auto-Relist immediately" (depends on duration I guess)?
Not suggesting you check the "The item qualifies for eBay relisting credit" or Inventory boxes, as those apparently aren't relevant to the decision to relist (relist 100% of the time). The relist either qualifies or doesn't qualify for the relist credit, but you get one if it does.
Still trying to figure-out Auto-Relist, and not trying to be a pest.
Danno