Have a look at https://discourse.pi-hole.net/t/dhcp-reservierung-wird-nicht-angewendet/19788/9:
Even if it's in German, the configuration file sample provided should allow you to work out a solution (might even have been a similar smart-DNS use case then). 
EDIT:
And here's another thought:
If you were able to pinpoint the domains that your Fire-TV stick is using for geo-locating you, you could try to just redirect resolution for those domains to your smart-DNS.
Add your domains to a custom dnsmasq file (e.g. /etc/dnsmasq.d/42-smartdns.conf) line by line, e.g.:
server=/<somegeolocationdomain.com>/<smartdns ip address>
Substitute the bracketed parts with your real values.
If that would work, it would allow you to stick with your router's DHCP server.
The tedious part would be to get hold of the correct domains, of course.