From how you described it I may assume that those BLOBs are totally irrelevant to user data, have no influence on user activity and may only be need for the system maintenance purposes.

If it's correct than the data logically does not belong to the database at all. And you don't need to back it up to any other instance, as the recorded errors are not relevant to that environment.

Quick fix could be creating a "app servicing" database attached to each production database and saving those blobs over there.

Minimal change to the code and immediate effect.

Then you may think if you want to have a backup plan for those databases and, if yes, what should it look like.