Example of the latter, if Pi-hole had RouterOS, you could do:
/routing table
add name=via-router fib
/ip route
add dst-address=0.0.0.0/0 gateway=192.168.88.1 routing-table=via-router
/ip firewall mangle
add chain=prerouting connection-state=new src-mac-address=AA:BB:CC:DD:EE:FF protocol=tcp dst-port=53 action=mark-connection new-connection-mark=from-router passthrough=no
add chain=prerouting connection-state=new src-mac-address=AA:BB:CC:DD:EE:FF protocol=udp dst-port=53 action=mark-connection new-connection-mark=from-router passthrough=no
add chain=output connection-mark=from-router action=mark-routing new-routing-mark=via-router passthrough=no
Where 192.168.88.1 is default gateway for LAN and AA:BB:CC:DD:EE:FF is its MAC address. But since Pi-hole doesn't have RouterOS, you'll need to find how to do the same with Linux.