If you're a roleplayer that's playing a lawful character such as an Imperial knight, then Jaree-Ra is likely to be an annoyance to you. On the one hand a lawful character wouldn't engage in a shady quest (Lights Out) where he's supposed to sabotage a lighthouse, sink an Imperial ship, and plunder it. On the other hand, there doesn't seem to be another path that brings Jaree-Ra to justice, and the prospect of seeing him yelling at you on the streets of Solitude for the rest of the game isn't particularly enticing.

How would you deal with the Jaree-Ra problem as a lawful character? Is there a lawful way to bring him to justice which I've simply missed, is there a mod that helps resolve the problem, or is there perhaps a different kind of approach?

It's unfortunate that you can't choose to report Jaree-Ra to the guards once you've gained incriminating evidence against him, thus sending him to jail and preventing the Imperial ship from being sunk.

Just ignore him.

I see him as anyone else I might come across in real life saying "hey buddy, wanna buy a watch?"

Assuming you are on PC:

Remove his essential tag via console + Frenzy without anyone noticing = Profit

Just pretend that the "stray" spell was an incident.

or better yet disable him and pretend like he never existed.

Can you remove his essential tag and give him a bounty? Or did they remove the ability for NPCs to have bounties after too many peasants were put to death over a loaf of bread in Oblivion?

Yep, pretty much this. I use this method for Jaree-Ra, Dervenin, Tyranus, etc,.. This is my preferred route to take with most annoying NPCs or quest ambushers that I don't want to deal with.

Like Brynjolf!