I got it to pop, finally. From Activision Support:
Sorry to hear your having a issue with the There is no Cake achievement. Also thank you for troubleshooting the issue before contacting us. When you were playing in Story Mode and collecting all the secondary objectives, did you finish the level before quitting out? Can you also please try deleting and re-downloading your Xbox Live Profile and check again? Note: When deleting the profile, select Delete Profile Only. Also, can you try competing and winning in another Infiltration map and another Capture the "Greased-Up Deaf Guy" match to see if the issue continues. Thank you.
I only had to win one Infiltration match for the achievement to pop (after deleting and re-downloading my profile).