I'm pretty sure I played "Tricks of a Trainer." The first question was "Which of these Pokémon can/cannot be found in the Hoenn region?" or something to that effect (I'm pretty sure it was the latter) and one of the answers was Pikachu (I remember this being correct, but because Pikachu is on the Hoenn Pokédex this would contradict my other memory.) Bulbapedia already records how a picture of Ash shows up when you get a question right, but he was also accompanied by the words "You got that right!"
Can you confirm whether or not it was Flash-based? There are a ton of Flash game sites out there with randomly-uploaded games, and there's always the possibility that lost ones are buried out there, misnamed or something.
This should probably be a priority, actually, because official support for Flash ends this year (grr...) and even though there are probably workarounds (use older browser versions, scripts that emulate Flash, etc.) these sites could be kicking the bucket soon.
I assume it was Flash-based given that most of the movie was Flash-based, but there was no way I would've noticed, especially when I was still in elementary school. It might've been posted somewhere else, but searching for the names of the games didn't give that many results when I did so.