I would like to speedrun a few challenge mode levels but there are no level speedruns, just the full challenge mode.

That's right. Actual levels are not done in speedruns, but by highest score. You can compete by going for as high a score as possible. Most world records are on youtube, and all challenge mode leaderboards are here: https://chr.mn/pik4/ The site is japanese but with chrome you can translate all webpages. The best way to compete is to simply upload runs to youtube and join the community.

Ok