I feel it's not really applicable here. Pihole has the advantage of funneling all DNS traffic (typically UDP/53) to a single endpoint and making decisions about the request.

A user using an LLM is probably talking directly to the service inside a TLS connection (TCP/443) so there's not a lot of room to inspect the prompt at the same layer a Pihole might (unless you MITM yourself).

I think OP has the right idea to approach this from the application layer in the browser where the contents of the page are available. But to me it feels like a stopgap, something that fixes a specific scenario (copy/pasted private data into a web browser form), and not a proper service-level solution some have proposed (swap PII at the endpoint, or have a client that pre-filters).