Hi crshinn,
Thank you for posting in Apple Support Communities! Do you get any type of error message when you attempt to set limits for the TikTok app? Are there any updates available for the app itself? You can confirm by going to the App Store, tapping on the profile picture at the top and scroll down to see a list of apps that need to be updated.
How to manually update apps on your Apple device - Apple Support
You should also confirm that the device doesn't have any software updates that need to be installed since that may cause features such as Screen Time to not function as they should. To check for any software updates that need to be installed by going to Settings > General > Software Updates.
Update your iPhone, iPad, or iPod touch - Apple Support
Hope that helps!