The title update achievements are actually much easier than they are supposed to be. At the main menu, choose Chill Mode, then choose the character you need (in this case, Frappe), then finish a run through all 5 stages. The final stage is the second ice area with the moon in the sky. When you make it to this stage, stop moving the controller, and it will end the run and bring up the retry menu and the achievement will unlock for completing time trial mode!
This is the character I had the hardest time unlocking. You have to hit 3 of the ice patches that make your controls opposite, but it has to be in one life. If you get 2 and then get hit once, even if you don't die, the counter starts over. Some runs, you won't even get 3 in. It's really frustrating, but it's definitely possible.
You can keep track of which characters you've completed this with by looking for a medal icon next to any character's name that's already completed it.
I tried to idle these but it failed the run. Make sure you move the controller every so often to keep this from happening.
Thanks to
imaidiot19 and
x BIG DOM x for helping clarify what the final stage looks like!