Good news – I easily knocked out the two drains/strainers and scraped the epoxy from them. Turns out the epoxy had no effect whatsoever on the sink – no bonding, no melting, no staining.
I used the putty as the staining isn’t an issue and all appears well. Got a complete seal around both drains and the excess was easily removed.
One more question: do I have to allow any cure time for the plumber’s putty? Since it remains pliable, I’m guessing not (especially since the vat doesn’t say anything about curing AT ALL).