The Bambu Labs mobile app doesn't work with LAN mode. I found this app for Android, "LanBu" which does a decent enough job. It's still in beta and wants to display ads. So far my PiHole has blocked all its ads so it's tolerable. I wouldn't mind paying for it, if it's a one-time free and it allowed access from outside my LAN.
If you can run a server at least for the duration of the print there is a home assistant plugin and a tool called bambuddy to monitor (but not slice and sent print job afaik)