The police do get involved, but it's gonna be a power struggle thing like with team Rocket, other than the "Lunar Beam" beacon, they have no proof it was Team Luna. The move is open for circulation WAYYY before Team Luna started using it, but certain rare TM stores and certain Pokemon have the natural ability to learn it (i.e. Lunatone, Clefairy, etc). Also Team Luna's version is slightly altered to where it will be visible for a long time afterwards (As opposed to a few minutes), up to all night.
And as for the law concerning the training, almost everyone in the Morea region has a job concerned with Pokemon. It's kind of a mastery of the subject kind of thing. I might change it to where her father is "making" her train Pokemon for her own safety or something like that.
Does the Lunar Beam idea have anything to do with the Dark Mark from Harry Potter? If not, then ignore me!! XD