The proper place to do it is onunload. onbeforenload is meant to be used when you may need to stop the user from leaving the page, like when there is unsaved data.
I would be weary of writing code to cover all cases because you're not willing to test all cases. If you do your research and find that's it's too complicated across browsers, yeah, go ahead and do as much as you can. However, I believe there is no problem with doing local storage onunload. I actually did run into a problem, trying to send information to the server onunload. That did not happen reliably across browsers so I did end up with an approach that used both unload and beforeunload, but only after due diligence.
So my suggestion is do it onunload, check all your browsers.