Is there anyway to request a new category for a game?

I wanted to request a battle tower speedrun category for Brilliant diamond shining pearl.

Ask in the game's forum or discord. Odds are, it likely won't be added.

That's kind of unfortunately because the other games in the series have a battle tower category, This one doesn't for some reason.

Can't speak for that game/series specifically but broadly speaking the most common reason for a category to not exist is simply that nobody actually wants to run it.