The half-pot / half-stack bet on the turn and the river check are both odd, and even more so as a sequence. Once you bet the turn, can you possibly fold getting 3:1 if he raises? No, so shove.

And given the actual turn action, since you're going to call when he puts you all in, get it in yourself. The problem here is that your choice of action on the turn left you with little or no fold equity on the river, which you should have realized before you bet the turn, and which gives you more reason to shove the turn while you still have more FE.