Assuming that 192.168.0.1 is your router (and DHCP server), your dig results show that your router doesn't know names at all.
Of course, Conditional Forwarding will not help in such circumstances.
This may well be the case for some routers, e.g. Ubiquiti's Edge Routers will only provide local hostnames once configured to use dnsmasq, or some Microtiks may benefit from a script accepted as solution in Conditional Forwarding + Mikrotik? - #12 by DanSchaper.
As any solution would be specific for the router model you are using, you should consider expanding your research to include your router's documentation and online support forums as well.