Thanks very much for all the great ideas! Since I seem to gravitate towards doing the hardest way possible, I think I will go with the idea of drilling out the hole for an 1/8" dia. dowel and then drilling a pilot hole. With an 1/8" dowel, no one will be able to see it. Judging by ebay and Amazon, they seem to be a crafts store item.
Some toothpicks and glue would be a good idea too.
Hadn't thought about a potential split in the core wood. In that case, a fatter screw would just make it worse.