These steps have been tested on Windows 10 and NVIDIA GPU (Titan Xp), so I can't 100% guarantee that these steps will work on an older Windows system or on a non-NVIDIA video card.

All you have to do, is manually run the Steam Overlay process with proper arguments. Run Tomb Raider: Legend and run SteamOverlayUI.exe with the following arguments:
Where gamepid is the PID of the process.

First of all, run Tomb Raider: Legend. Once you reach the Main Menu, press ALT Key + Tab Key together to minimize the game.

Now, open the Windows Task Manager by pressing CTRL Key + Shift Key + ESC Key together.

In the "Details" or "Processes" tab (depending on your version of Windows) find "trl.exe" (without quotes) and read on the "PID" column what it says. Let's say that gamepid is the PID.

Now open the directory in which your Steam.exe is located. In order to find it, go back to the Task Manager, right click Steam.exe and click "Open file path".

Once you're in your Steam.exe directory, right click GameOverlayUI.exe or GameOverlayUI (depending on whether your Windows Explorer has the option to show file extensions enabled or not) while holding the SHIFT key (IMPORTANT), and click on "Copy as path".

Now press Windows Key + R Key to open the "Run" dialog box. Paste the path in the textbox. If the path is not between quotes, please add a quote in the start and a quote in the end of the path. Let's say that overlayuipath is the path.

Edit the command in your "Run" dialog box so it looks like:
For example, it will look like this in my system if the PID is 7944 and SteamOverlayUI.exe path is "C:\Program Files (x86)\Steam\GameOverlayUI.exe":
Once the right command has been written, click on OK and go back to the game.

Press Shift + Tab several times and you'll see that the Overlay is now working (and the screenshotting feature as well).

This is a problem that doesn't always happen, but sometimes it does. If that happens for you, proceed as explained below:

• Kill GameOverlayUI.exe from the Task Manager.
  
• Close the game.
  
• Do the steps as explained in the previous sections, but use the following command line to run GameOverlayUI.exe instead:
  STEAM_PID = PID of Steam.exe process
  GAME_PID = PID of trl.exe process
  
  GameOverlayUI.exe -steampid STEAM_PID -pid GAME_PID -gameid 7000 -manuallyclearframes 0

Everything should now work as expected.