If Portal 2 has been updated, but this mod has not, then two bin files in the mod "Thinking with Time Machine" are incompatible with Portal 2, hence the crash at the start of the new game.

The mod "Portal Stories: Mel" works fine for me, but this mod and the mod "Aperture Tag The Paint Gun Testing Initiative" crash for this reason.

• First, make sure to have Portal 2 and "Thinking with Time Machine" installed on the same hard drive
  
• Next, locate the Portal 2 and "Thinking with Time Machine" game files, you can find both with:

Right-click on the game in Steam Library -> Properties -> Installed files -> Browse...
or
Left-click on the game in Steam Library -> left-click the settings icon on the right hand side -> Manage -> Browse local files

First bin folder to replace:

• in "\steamapps\common" go to "\Portal 2" and copy the bin folder. Next in "\steamapps\common" go to "\Thinking with Time Machine" and paste the bin folder, select "Replace the files in the destination"

Second bin folder to replace:

• in "\steamapps\common" go to "\Portal 2\portal2" and copy the bin folder. Next in "\steamapps\common" go to "\Thinking with Time Machine\TWTM" and paste the bin folder, select "Replace the files in the destination"

You do not need to paste anything to "\Thinking with Time Machine\portal2"

Warning if you do "Verify integrity of game files" after the above fix, steam will replace the updated bin files with the outdated ones. Hence, you will need to replace the two bin files again.

The game should work now, but if not, here is an old post by the developer:

https://cs2bus.com/app/286080/discussions/2/34096318970765001/

The mod "Aperture Tag: The Paint Gun Testing Initiative" community hub might have useful solutions:

https://cs2bus.com/app/280740/discussions/

or this:

https://www.pcgamingwiki.com/wiki/Aperture_Tag:_The_Paint_Gun_Testing_Initiative#Issues_fixed