This is the second pet battle where it will not complete, I have beaten the pet opponents, but the ? remains silver, I’ve restarted the game and done the quest again and still no change so just had to completely abandon the quest and forget about it.
It sometimes happens - mostly when you have first battled Erris or after you lose or restart the match for any reason.
If you do it on a different character you’ll find the ability to hand the quest in after winning resets.