When playing Halo 4’s Spartan Ops mode, DLC cosmetics (such as the Mark V armor and DMR MCL skin) are replaced with the Recruit armor and default weapon skin. I’m not sure if this is an issue when being played on actual Xbox 360 hardware, but it happens with the Xbox One’s Xbox 360 emulator.
This is a well known and legacy issue OP. Due to the way that DLC items work in Halo 4 it wasn’t possible for them to get these items into Spartan OP’s. Unfortunately not something that can be fixed
Referring to the other part of your post, it happens when playing on 360 and One both. As I recall, the reason was due to how the engine handles skins. Adding the new skins into Spartan Ops would require more resources than were available from my understanding. It would have been very cool to be able to use the DLC skins in SpOps for sure.