Wich SC version are you running?
Is the FCI instance owned by a non-cluster owner?
i am asking because If the FCI instance is owned by a non-Cluster owner node, backups can fail with the error:
"No SQL resource is discovered on the host. Check SQL resource(s) state on the host."
If the FCI instance is moved to the Cluster owner node, the backup will complete. This also happen if the user under which the backup is running does not have permission on the cluster instance.T he log-in user or the group which the user is part of has to have to be part of the local administrator group of the "Cluster owner node" for
backup to work. Typically, sysadmin user may not be part of local admin group.
I've seen this issue in SC 2.0 ans should be fixed in the next relases (starting from SC 2.0P1)