So this should be unmissable and no matter the ending you have, it should pop however there is a glitch that was even present on the PS5 version that I encountered which meant it didn't pop (same goes for all trophies that were meant to pop at this time). The ONLY solution that worked for me after doing 7 full play-throughs of the DLC is as follows;
- Reset the PS4
- Reinstall the game (DO NOT START THE GAME)
- Try and stop the patch from applying to the base game (I don't think it's needed but I stopped it anyway)
- Install Sam's Story DLC
- Disconnect the PS4 from using the internet in Network settings and launch the game.
- Then play through all of Sam's Story and it should everything popped.
- Then re-sync your profile when you've turned the internet back on and wait about 5 minutes and it should show as done.
Hope that helps, good luck!