IW4.X neither runs with the steam overlay, nor it appears as a game in the friend list anymore. But hear me out, I found a very ubiquitous command to trick steam into thinking that I'm playing the multiplayer application of MW2, but in reality I play IW4.X!

First step is having downloaded at least the multiplayer variant of MW2 and have IW4 X fully downloaded.
Assuming the mod is downloaded in the game path's directory, in which the two executables are located, right-click to it and navigate to Properties...>General>Launch Options. You'll come to the point where you see this:

Then type in the textline the following command:


Where you see [YOUR GAME PATH] literally means the path your game is being stored, acording to the File Explorer. It looks something like this:
C:\Program Files (x86)\Steam\...
*You could find the exact file path of your game in Manage>Browse local files

After that you can run the multiplayer of MW2 and you'll be greeted by IW4.X!! (and its sketchy command prompt)
Now you can play the mod as if you're playing the original multiplayer. You can pretty much do this with any program in your computer, it doesn't matter. Now you can rack up some hours that get recorded on steam and enjoy this epic game!