Had this thought the other day while my girls were watching me turn. They wanted "Wooden Fish" too. But theirs had to have real tails made from wood as well as flat sides. "You know, like real fish Daddy"
Anyway, the way I found to do the tails thought might help do a popper mouth. I turned the body then when I got to the tail I left a large flair at the live center side. Looked like a polaris popper profile. Well the tail could and would be cut with the scroll saw from there but I decided to take one more step on the lathe and bowl cut the flaired end. When I removed it from the lathe I had a polaris popper! Only thing left was the "Tang" or "nub" that the center was holding. We're talking about 1/4" dia x 7/16" high max. Easy to dremel that out. Like I said that ended up the rough "wood tail" on mine. I cut away the sides of the flair dia and up the sides of the plug and had my tail. When I move on to poppers I think I'll start out this way and see how it goes.
Just an idea...