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)

Relevant cards

Note: All applicable cards may not be listed below.
SM11_190
SM11_190
SM11_232
SM11_232
SM6_55
SMA_SV19
SMA_SV19