For anyone interested, you can automate the process of searching and killing the handle in Process Explorer, though you need the command-line version of it called Handle, it's from the same person who made Process Explorer and can be found googling Handle Microsoft.
To find the process, you run something like this:
handle -p D2R.exe -a "BaseNamedObjects\DiabloII Check For Other Instances" | findstr /C:"D2R.exe" | findstr /C:"Sessions"
which outputs:
D2R.exe pid: 17600 type: Event 550: \Sessions\1\BaseNamedObjects\DiabloII Check For Other Instances
Then you grab the PID (17600 in this case) and handle hex (550 in this case) and then you can run this to close the handle:
handle -p 17600 -c 550 -y
If you're good with coding, you can set it up so you just double-click the .bat script and it automatically kills the handle.