there is some room for error with the shimano 3 speed.
the 2nd gear is top dead center lets say. (as indicated here)
this has to be exact in the neutral position. (2nd gear)
release all tension and your gonna be in first (or 3rd) sorry brain farting
pull it as far as it will go and you'll be in the remaining gear (first /third)
that being said, you dont have to release all tension before it changes gears, and the cable doesn't have to be totally tight to switch into the higer gear. there is room for error.
now if you add a heavy spring to your heal/toe shifter, with the cable attached to the spring, all you have to do is figure out 2 gears. top dead center(neutral) and no tension on the cable.
for the 3rd gear, the spring should compensate for any extra travel after the cable is fully taunt.
a heal toe would totally work as long as theres something to lock it down in each gear.
i'd mount a plate to the pivot section of the heal toe.
drill 3 holes to match the position of the shifter in each of the gears.
then you just need a spring powered ball bearing(mounted to the frame) to index each of the holes