Hello Tmstro,
Does it seem that all of the iMessage apps work as normal for your daughter when Screen Time is disabled? Testing that out can help to confirm if the two are connected, if you haven't just yet.
This page can assist with providing some helpful info on using parental controls and what can be restricted via Screen Time: Use parental controls on your child's iPhone, iPad, and iPod touch - Apple Support
• Music, Podcasts, News, Fitness: Prevent the playback of music, music videos, podcasts, news, and workouts containing explicit content
• Music Videos: Prevent finding and viewing music videos
• Music Profiles: Prevent sharing what you're listening to with friends and seeing what they're listening to
• Movies: Prevent movies with specific ratings
• TV Shows: Prevent TV shows with specific ratings
• Books: Prevent content with specific ratings
• Apps: Prevent apps with specific ratings
• App Clips: Prevent app clips with specific ratings
Cheers.