Pi-hole's DHCP service should announce different DNS to clients

This is not the expected behavior. The expected behavior is that Pi-Hole provides itself as DHCP to clients, so they will route DNS through Pi-Hole.

There is a method to use an alternate DNS for individual clients, but this is not the default Pi-Hole behavior.

To do this, Pi-Hole must be the DHCP server. These thread provide guidance.