Thanks to GOG user sifaaa for their post over on the GOG forum here. I have done some testing and found some additional steps required to get it functioning.
Copy these three files from the base game Diablo folder to the dx sub-folder:
BATTLE.snp, bncache.dat and bnupdate.exe
open up Notepad and paste in the following (and fix the game path to your actual game path):
REG ADD “HKEY_CURRENT_USER\Software\Battle.net\Configuration” /v “Battle.net gateways” /t REG_MULTI_SZ /d “1000\005\0uswest.battle.net\08\0U.S. West\0useast.battle.net\06\0U.S. East\0asia.battle.net\0-9\0Asia\0europe.battle.net\0-1\0Europe\0connect-forever.classic.blizzard.com\08\0Globa l d1\0” /f
and save the file as Diablo.bat somewhere (Your desktop is fine)
Add \dx\Diablo.exe to your windows firewall exceptions (or edit the one GOG provides for the original exe)
Forward ports for Diablo as described here.
Here are the results:
The Battle.net interface is a bit broken, but it works. Click on the text outside the window, not the actual in game buttons!