Cadmus was already destroyed, check the timeline.
He could certainly try, but without access to his ring he has no way to beat the leagues security.
Only the local equivalent to the national enquirer would pick that up.
Not really, he would need really hard proof before the courts wouldn't just toss the claim out.
Even if Cadmus was destroyed, governments would still be wary of the League, but now they would hesitate before sending an army of metahumans to attack them, but taking legal action against them may be something they are wiling to do.
He may not have his ring, but in case you've forgotten, which I'm sure you have, he has at least dozens of scientifically gifted people working for him, who may be able to beat League security.
The Leagues security has been beaten before, and it didn't require a power ring.
Other newspapers could decide to pick it up, due to the fact that it is an interesting story with great ramifications.
And Peter could also just buy newspaper companies, respectable ones, and have them publish it.
Peter could easily be recording his meeting with the League, where they admitted they stole his ring and gave it to someone else, using some of his advanced tech, that the League can't detect due to its nature.
The courts would also be hesitant to just toss away the case of the guy that's supplying multiple countries with electricity, and who has a small army of supervillains working for him.
They may be okay with the League punching ordinary crooks in the face and arresting them, but they may draw the line at the League breaking into the homes of productive, and wealthy, members of society and stealing their property without good cause.
The League in Earth 12 has no charter with the UN that allows then to do this, or any charter really.
If you say that they do have a charter then leave actual evidence like a link to it, because your word is shit.
Peter and the courts could also ask that Wonder Woman uses her Lasso on both Peter and Batman to see if the ring was really stolen, and if both she and Bats refuse that, well, then people will become suspicious of them.