I don't know of a way to get a string directly from the game onto the Windows Clipboard, so any solution is going to involve some form of manual copy-and-paste. A while back I wrote a little program that watches the log file for a special line containing the URL and opens the external browser, but it was a lot of hassle for very little gain. Maybe I should look again at it.
Even though it's awkward, the most efficient method I have found is:
Click on the URL in the in-game browser
[Ctrl] + A (highlights the entire line)
[Ctrl] + C (copy to clipboard)
[Ctrl] + V (paste into external browser)
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