SQL Cluster instance does not see all drives
I believe that is the whole purpose of FCI. You will only see the disk part of shared storage in that cluster.
I am not sure what you are trying to exactly do when you say local backup. Backup of database ? If yes then I guess that should be simple enough to backup in the drive part of shared storage so that after failover to other node backup continues to work.
There are few drives which we might not care local installation directories to be part of same shared storage or software drives or dba script drives what we have in our env.
So totally depends what drives are you talking about here