<< Apparently, the strong acid in the stuff worked them into the cement. >>
Stomach contents may contain a small amount of acid, like HCl. So yes, that can
attack concrete. Try treating a small area with dilute HCl (muriatic acid). If
that doesn't work, switch to something highly alkaline, like TSP. For more
choices consider looking at products offered in janitor supply outlets. The
pros often have unusual items not commonly found at the box stores.
<< We'd like to avoid painting. >>
Why? That may by far your best option if nothing else works. By the time you
give up the struggle with the stain the area should be quite paint receptive.
Whatever, good luck.