In which timeline are you? Harambe alive or Harambe dead?

Why not? The double jeopardy clause is about prosecuting a single person twice; it says nothing about prosecuting a second person for the same crime. Heck, convicting a second person wouldn't even automatically invalidate the first conviction. (SCOTUS has ruled that innocence is not a sufficient reason to overturn a conviction.)

Remember, we have a judicial system. Calling it a "justice system" is inaccurate.

I think he wanted to be caught and make a statement. "Those parasites had it coming"

The one person interviewed by media who had spoken with Luigi said the note (manifesto/confession) wasn't at the standard of Luigi's other writing, his style, his philosophy, his reading

The thing is, whoever did it was pretty smart about much of the crime, which is why the story of how Luigi was caught and what they say he had on is person seems so odd relative to the behavior the previous week.

If it was Luigi, he was pretty disciplined and smart before, during and immediately after the crime with the only leak being showing his face at the hostel (which he may have been required to do when he gave them ID).

If he did do it and really had all that evidence on him, maybe he wanted to get caught? It seems crazy that anyone would hang onto a 3d printed gun when it would be pretty easy to destroy and dispose of it in multiple pieces and locations.

I’ve said it before but I truly believe that Luigi would still be free if he had plucked his eyebrows, regardless of whether he did it.

Edit to add, maybe he thought he’d get shot or captured at the scene and didn’t think too far beyond the immediate getaway? And with the stress of his picture up all over the news and social media, he kinda wasn’t thinking straight?

A slight variation on your second scenario is parallel construction.
I don't know if Luigi's the shooter or not, but if he did it, there's a decent chance whatever is presented as evidence has absolutely nothing to do with how they actually caught the guy.

Luigi did it, and he really was that stupid.

Given that he was valedictorian at his school, I find that unlikely to be the option.

I think you're missing a 4th option:

• He did it, and believed it not worth the anxiety/suffering of living the rest of his life as a wanted man, so did what he could to get away from the immediate scene but ultimately turn himself in so to speak.

And there are almost certainly more nuanced options out there.

How Bazaar

That would explain that part that says

PS: "Oh yeah that Agent Smith guy who called in this arrest, he needs to be promoted and his co-workers need to stop complaining about his body odor, because it's quite pleasant actually. Prefer it to an air freshener."

I think that the gun and everything else were dumped and found and the cops are saying he had them on him when he didn't. I think they know it's him, but they didn't use legal means to identify him (parallel construction). They might have used some kind of new way of tracking people that they don't want to reveal to the public.