I think you should have a look at what your system does to /etc/resolve.conf,
or more specifically, how it builds it. Look here:

/lib/dhcpcd/dhcpcd-hooks/20-resolv.conf

Also look at the man page for 'resolvconf" to get an idea of what is going on.
Also look at the dhcpcd man page.

I just cannot remember off hand if dhcpcd will take domain info from the
dhcp server that provides the lease.