If I used Poipole's "Knockout Reviver" attack, but then on my opponent's next turn they play Channeler, would that allow them to take a prize for KO'ing Poipole?
No, it would not. The effect of Knockout Reviver resides on Poipole, and the opponent's Channeler only clears effects on the opponent or their Pokemon.
Source: TPCi Rules Team (2019-08-22)