Matt, thank you for the pics! Nice setup you have there. Very helpful info, much appreciated.
I have bought that same check valve. I was wondering if I need a P trap (or S trap).
Also, I like that you can see the liquid in the tub so you can see where its level is.

the pump I have is Liberty pump 403. Bought it off craigslist. It is sitting right next to my sink and I connected the sink drain right to
the pump inlet. There is an S trap in the PVC before it connects to the pump. Should I keep the S trap?
Next I'll connect PVC to the discharge outlet and up the wall to the house drain. WHich is a whole other adventure to figure out connecting to that.