Running it into the bottom of the boat and pumping it out with the bilge doesn’t sound like that great of an idea to me. Just not sure I’d want all that fish slime rotting in the bottom of the boat on 90° days all summer.
Mine (factory) has two pumps… one fills from the outside on a thru-hull connector, the other is a submersible inside the well that drains it. It does a pretty good job, and after I empty it I run a 5 gallon bucket of tapwater through it and that’s enough to keep it from stinking/plugging the pump. If I were to install one, I would consider this setup because there wouldn’t be any underwater fittings inside the livewell that you’d have to worry about sealing. Also, the drain pump has a T with two valves so that it can be used as an aerator/recirc (even when the boat is on the trailer on the way home, which is pretty slick).
If it’s as nice this weekend as it’s supposed to be, I may uncover it and put the troller on. If I remember I’ll take a picture.