Normally, adding Syncthing to “ignore battery optimisations” in Settings should be enough. If there are still problems even with that, you can use brute force to set the screen to always stay on while charging in the Developer Options. All this assuming stock(ish) Android, since different vendors seem to like implementing a variety of other methods to prevent apps from running in background.
Please open the Web GUI on the phone (or remotely), enable config debug logging, and then pause and resume the affected folders. After that, open the log and look for lines similar to
DEBUG: Detecting FS at "/sdcard/test" on android: Setting mtime window to 2s: usage.Fstype == ""
If you find them, please copy and paste them here.