You inadvertently stumbled on to the solution by installing stretch, since the fix is in dnsmasq 2.76.
The actual problem is detailed below, along with a fix for jessie.
The solution I first discovered here:
https://www.raspberrypi.org/forums/viewtopic.php?t=128449#p919942
Then found the bug ticket here:
https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=860064
The diff patch to fix is here:
https://bugs.debian.org/cgi-bin/bugreport.cgi?att=3;bug=858506;filename=dnsmasq-init.patch;msg=27
You can install a more recent version of DNSMASQ here: