I just solved a similar issue, not so much regarding permissions but when you chroot bind to a new directory such as /var/lib/named, then bind considers /var/lib/named as it's new root home directory.
In otherwords, in the config files if you set /var/named/.... for any files, it ignores the /var/lib/named/ as it considers that it's root. So any new configuration files or the named.pid file must all be in the same directory structure within /var/lib/named.
My issue was getting an error: "couldn't open pid file '/var/run/bind/run/named.pid': No such file or directory" - yet that directory existed in the literal structure of /var/run/bind/run but NOT in /var/lib/named/var/run/bind/run
Once i created this structure within /var/lib/named and set the permissions, all was right with the world again.
I hope this helps others.