Well, it has taken a while for me to get back to this. I had unrelated problems with my WiFi that were more urgent and which prevented testing.
This is now working. In the DHCP Lease Table on the router, the WiFi is now listed using the user-specified string. (I don’t know whether something changed in the operating system or I stuffed up the change or something changed in the router or combination thereof.)
Unfortunately I have changed both dhcp-hostname and dhcp-client-id (to the same string) so I don’t know which one worked.
So … tying this back to the original topic … yes, if you have a portable device that you use with untrusted access points, you should override the default behavior in this regard.
If you are randomizing the hostname then it probably doesn’t matter too much, but the truly paranoid would also randomize the hostname sent in DHCP requests and ensure that the overridden string is sent (rather than the system hostname) or ensure that no hostname is sent in DHCP requests.