I'm looking for a small inkjet printer / ink combination that I can use to print a small logo and some text and address on the back of 4x6 postcards in very short runs of 100 per month.

The postcards are offset printed on the front side. The back is not Aqueous or UV coated, but it's a standard slightly glosssy slick coated-for-offset 16 pt. base stock - not porus to accept inkjet ink.

Would an Epson DURABrite® Ultra pigment ink set on this?

The straight paper path should feed nicely without leaving them curled, but dry time might still be a major issue even with epson's pigment ink. (the old dye printers will definitely not work.)

How about this:
Take off the small epson paper tray
Get a 6' or 8' standalone outfeed conveyor belt used. Something well used and cheap. You might have to rig the pulley to run it super slow for the slow inkjet speed to give maximum dry time on the belt.
Rig up a heater / blower above it blowing down on the cards as they ride the conveyor.

Maybe a crazy thought, but I think it would work to dry the inkjet ink and prevent smudging.

Just a followup -- the durabrite ink "sort of" works. The C M and Y dries almost instantly, but the black ink takes about 3 hours to fully dry so it's smudge proof. I'm not sure if the ultrachrome in the bigger units would work any better, but they're a little too expensive to buy just to test with like I did with this durabrite ink.