Could do a DIY solution with Syncthing’s REST API (https://docs.syncthing.net/rest/system-pause-post.html), or switch from SyncTrayzor (which seems to be a dormant project) to another Syncthing wrapper that’s actively maintained and has built-in support for pausing via CLI: https://forum.syncthing.net/t/yet-another-syncthing-tray/8502.
Whenever Syncthing is paused, it’ll pick up where it left off when it’s unpaused, so other than files being out of sync while the pause is in effect, it’s not going to break anything if that’s what you’re wondering.