You could just serve a webpage that queries your desktop, and visit that with the browser on your phone. Slightly more work, but definitely viable.
You can even make a shortcut on the home screen of your phone that automatically navigates to your custom search page.
Do alternative iOS browsers allow exchanging the search engine? As far as I'm aware Apple only locks them into using WebKit.
I switched to duckduckgo on my phone for now.
I'm thinking about building a search page where the query goes to my home desktop computer where I run headless Chrome to scrape results from all the search engines and then forward them back to my phone. Unfortunately, Apple will not fucking let me use a custom search engine in Safari on the iPhone, probably because they're in cahoots with Google and China to bring about a new world order where nobody has any freedom whatsoever. xD