Is there some kind of known issue with the Debian Jessie configuration used in the Turnkey appliances that prevent them from properly registering DNS with dnsmasq?

For example, my router runs dnsmasq and provices both DHCP and local DNS resolution services for my network.  All of my non-Debian Jessie clients (Ubuntu, Windows, Alpine, etc.) all properly register their hostnames when they request a DHCP lease.

But none of the Debian Jessie clients do, including V 14.2 of Turnkey Core that I just downloaded last night and loaded up in KVM.

I don't claim to be an expert on Linux networking, so if I'm barking up the completely wrong tree here, please let me know, but one of the things I found different between Debian and Ubuntu is that my Ubuntu clients have a /etc/dhcp/dhclient.conf file with this (among other contents) line: send host-name = gethostname(); - but on Debian, that file is missing completely.

Would the absence of that file cause this issue?  Or is there something else going on?

I also took a look at the issue tracker and found Confconsole - Hostname - does not set FQDN and hostname at once - but I don't know if that issue and what I'm encountering are related, especially because I've manually updated the /etc/hosts file on my Debian clients and rebooted, and am still encountering the same issue - they obtain a DHCP IP, but do not register their hostname with DNS.