What's the best method to get a loose Tuttle fin to fit perfectly. Shim, tape, epoxy? What tape, what epoxy? Epoxy a better permanent fix?

Depends how loose, a couple of coats of spraypaint may do it (but not really durable)

If u need thicker:
clean very well
sand real rough with 40grit\
clean again
brush with epoxy both sides

slowly block sand to get tight fit.

it may still flake off later if the coat is really thin in some places.

I use mylar strips (sail repair tape) on mine. cut to size with the corners rounded so it doesn't peel.

If if it's really loose, might have to add some filler as Mark said and re-sand it to size.