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All methods I or Google found to move directories in a .tar file requires me to extract the files and then place them back in. The problem is, I don't have access to the account or any account in the group that the directory's permissions in the .tar file specifies, so I can't extract it with the exact same permissions, but I still don't want the permissions for the directory in the .tar file to change. Is there a way to move directories in a .tar file without breaking permissions (thus definitely without extracting the directories first)?

Just do it on a machine that you have root on. Copy the files as needed with the preserve attributes flag (-p) and then retar. That'll keep everything as it should be.