I feel like I'm missing something here...
Who's going to be fingerprinting DHCP messages on your home network?
Outside of that, fingerprinting or tracking any DHCP info would be the least of my concerns. You have 0 control over any data the moment your devices connect to a public network. What use is DHCP info when you can person-in-the middle all the traffic anyway?
And anyway, what info are you concerned about? Having had a VERY quick browse of RFC2131 the worst thing would be "leaking" the device MAC address which can be discovered via several other means anyway
Exactly this, yes