The problem with old inner tubes is that they degrade quickly. I use them to cover my seat clamp area to prevent egress of water & grime into the seat tube. But they only last a few years.

Since carbon frames are built with epoxy resin, I don't know why you wouldn't use the highest strength epoxy you can find. Better yet, contact the frame manufacturer and ask them what adhesive they used. (Then again, it failed, so there's that.) THEN I would consider wrapping the area with durable tape. I wouldn't go without the plate.