What’s the best system for running an Ace Attorney style legal battle (complete with over-the-top shenanigans)?
I’m tempted to say Sea Dracula, but that would be a lie - Sea Dracula fosters a very different sort of shenanigans from Ace Attorney.
(Though if you’re curious what a game of courtroom drama that uses dance-offs as its conflict resolution system - yes, as in the players physically get up and dance, while any uninvolved parties act as judges - you should check it out. It’s only a dollar.)
Honestly, if you’re looking for something rules light I’d probably just run it in Fate Accelerated Edition. FAE is good for most premises where all of the player characters have basically identical descriptive capabilities and success and failure typically boil down to how their personalities bounce off of each other. It doesn’t really have stats - characters instead have ratings in “Approaches”, like Forceful, Flashy or Sneaky - and it doesn’t distinguish between physical and emotional injuries, so you wouldn’t need separate sets of traits for the investigative and courtroom bits.
For a rules-medium option, you might instead have a look at Skulduggery. Its primary conflict-resolution mechanism is arguing, to the extent that the rules for physical combat are considered optional, so courtroom drama is a natural fit. Plus, its temptation mechanics could be adapted to give mechanical weight to the sorts of eccentric personalities you run into in Ace Attorney. In a nutshell, each campaign has a defined set of opportunities to be terrible - which can range from stuff like Pride and Lust in grittier games, to stuff like Pedantry and Spite in lighter ones - that charaters must roll to resist whenever a clear opportunity to indulge them arises. Which means that in the right sort of game, you can totally stat up a character who has to save vs. nitpicking whenever people are wrong on their presence. A potential complication is that the game’s default assumption is that it’s about awful people being awful to each other, so you’ll need to keep a careful eye on how you’re statting things to make characters who aren’t total jerks work.
Now, if you wanted to go rules heavy, Ace Attorney basically views courtroom drama via the lens of shounen fight manga, so there’s part of me that’d be terribly tempted to do it as an extremely variant setting for Anima Prime. The manoeuvre/strike rhythm would be an excellent fit for the dynamics of the courtroom scenes. Of course, you’d need to come up with a completely new list of skills and powers, which is why this last one isn’t a serious recommendation!