Update: This fix may not be usable on Windows 10 as is and need a "mini image" mounted in a virtual drive. Check Gamecopyworld for that if needed.

Info is taken from the text file in the mod.

This may not work for everyone, but it worked for me. It requires you to already own a different GFWL game on your PC, and/or install the client from http://go.microsoft.com/fwlink/?LinkID=201134 as it doesn't include the client itself in the mod (which is only about 5.5MB itself).
GETTING THIS TO WORK REQUIRES THAT YOU USE YOUR SERIAL KEY FROM A DIFFERENT EARLY RELEASE (NON-SSA) GFWL GAME!

From the included readme (with a couple of edits):

To restore Steam (and probably GOG) version of Blazblue online functionality, you will need another different GFWL game, as a cd-key isn't supplied with the newest version.

1. Right-click the game in Steam and choose properties. On the "Local Files" tab, press the "Browse Local Files" button to open the game directory in the Steam folder.

2. Copy the "1st Place in Blazblue directory with BBCT files" archive into the game folder and run it, selecting yes to overwrite when prompted.

3. There is now an empty folder in the game directory called "BBCT". MOVE all of the game's files into this new BBCT directory so there is nothing left in the base game directory except the BBCT folder (now containing the game files) itself.

4. Next put the "2nd Extract in newly empty Game directory that only has BBCT folder with game files in it" file in the directory next to (NOT INSIDE) the BBCT folder and run to extract it. You will now have a new bbct.exe, bbct.bat, and command.txt file next to the BBCT folder.















5. Run Blazblue in Steam as normal, which will now bring up the restored game launcher menu. You can edit your controller, and press "Launch" to load the game itself. When prompted by GFWL, enter your GFWL user name and password. On the following screen when it asks for the cd-key, just enter any key from one of your other GFWL games. If it doesn't work on the first attempt, try another one of your GFWL game cd-keys.

6. You should now be signed in to GFWL, and have restored online VS., Replays, and Achievements. The Steam overlay will not function. If you want an overlay and in-game Steam chat, use can use a 3rd party one such as Evolve or Raptr and it should work.

7. To go back to the normal Steam version, first MOVE all the files from the "BBCT" folder back into the normal "BlazBlue Calamity Trigger" folder. Then, in the Steam properties window for Blazblue, again on the "Local Files" tab, press the "Verify Integrity of Game Cache" button, and Steam will check and replace the game files with the standard Steam version ones.

mod file: https://www.dropbox.com/s/lqu1n97xtlz6dtq/BlazBlue%20GFWL%20online%20matchmaking%20restoration%20for%20Steam.rar?dl=0