My I suggest rather than updating /etc/resolv.conf a static value should be placed in /etc/systemd/resolvconf.conf.  This way regardless of network interface it will maintain the value.

I have also seen cases where the DNS IP does not get updated correctly after reboot on Pi3 systems when using DHCP.  From the best I can tell, it appears the correct values are written to /var/run/resolvconf/interfaces/wlan0 but resolvconfig tries to read it immediately after but it was not written yet (slow flash card).  My recommendation to work around it to add $(which resolvconf) -u to /etc/rc.local.  This give enough time between the write to the flash card.  The result is /etc/resolv.conf has the correct values.

Your thoughts?

Darren
KJ6KFB