You are in a Catch-22 here.
A slow-drying, solvent-based paint marker that lays down a thick heavy layer of paint will look great, and if it eats into the body of the plastic, even better, right? because the (embedded) signature will survive even if the pigment sloughs off.
On the other hand, a good fresh Sharpie that stains down into the plastic just a bit, but dries fast, will be far more forgiving.
My advice? Trial and error. Get a couple of garbage boards at a thrift store for $1 each, and experiment with a fresh Sharpie, waiting "x" seconds (do it in 10-second increments) and wiping with a dry paper towel and/or a wet sponge to see exactly how it reacts.
That way, you should get a feel for exactly what to use and how much time you should expend on each encounter.