Headfirst is right. Seconds pass as Hawlucha doesn't twitch a single feather in any sort of attempt to escape from the simultaneously compromising and embarrassing predicament. Shay's Rotom floats over to be extra sure.

"... Hawlucha is unable to battle! Victory goes to Popplio!"

With that confirmed, Shay nods without argument and pulls out Hawlucha's pokeball, recalling his bird back into it.