Here is a workaround for the bug/glitch, the team is aware of this is happening and currently working on it.
For PC you can try this:
1. Rename the settings folder and disable cloud save
2. Play the War Story, completing the challenges that you were previously missing
3. Receive the Mission Reward
4. Rename settings folder again - the skin will remain in the inventory once claimed.
For PS4 you can try this:
1. Export the game save files to an USB drive.
2. Delete the game save files in the console.
3. Start the game. You'll have to adjust contrast and play again the intro.
4. Now, play the War Story and complete the bugged challenge. It will show correctly achieved.
5. Close the game, and restore the game save files from the USB to the console.
6. Start the game, and you'll have your reward at the armory.
For Xbox it would similar:
1.You would need to move the saved file from the Xbox to a USB stick,
2. Clear local and cloud saves for BFV. (This wont remove multi player data)
3. Start the game. You'll have to adjust contrast and play again the intro.
4. Now, play the War Story and complete the bugged challenge. It will show correctly achieved.
5. Lose the game, and restore the game save files from the USB to the console.
5. Start the game, and you'll have your reward at the armoury.