I think I came up with a solution.
I downloaded a copy of Mozilla SeaMonkey and tried to paste in the above URL and it worked!
Now I still had the problem of that you have to manually paste it into SeaMonkey.
I then searched and searched and found this tool called "More Internet":
http://www.monkeyfood.com/software/moreInternet/ (I had to use the alternate download link at the bottom of the page)
It allowed me to configure SeaMonkey to be the default handler of any URL that begins with the "mailbox://" protocol.
And that did it!
Now when I drag an email from Thunderbird to another application where it is rendered as a URL, I can click on that URL to bring up the email in SeaMonkey.
I would still like a solution that forces Thunderbird to handle this link, but this is good enough for now.