Well to me sounds strange because:
1- The ship is two days away from Portsmouth, fully loaded, That’s , let’s see… two days lost+ two days to be back+ lets say 1 day for repairs+ two days to be back to the point where the ship was. One week delay + fuel expenses etc. That is a lot of money!!
2- Illustrious is the flag ship of the task force, so the rest of the task force gets delayed for a week
3- It does not matter how big the fridge is, the room itself does not break down!!, the cooling parts do, coils, compressors, evaporators etc and they are not big parts (relatively )nor is the technology or skills required to fix them difficult to find or supply
4- The task force is going to be doing numerous stops around the Med. How difficult it would be to fix the fridge during one of this med stops. As you said, the ship is not half way around the world!!, they do not need to fly anything, they could even send it by lorry!!
Regards