Rewrites - can we use this to replace Safari's built-in search

I have created an instance of SearXNG (a personal/private search aggregator) and deployed it on my Tailscale tail-net as its own machine. This means it has its own, Tailscale IP address, URL and https certificate.

I primarily use Safari on both my Mac and iOS devices. Although you can choose which search engine to use, the choices are limited. I was hoping to be able to select Yahoo (search.yahoo.com), then use a NextDNS Rewrite to map "search.yahoo.com" to either the IP or URL of my personal search instance.

I've tried this, but am not getting any luck. Does anyone have any suggestions on this?