GCP - my options for hosting other than Firebase

Question

I have a web app which connect to a Cloud Function (not Firebase, but GCP). The cloud Function is Python, it is massive and it connect to SQL on Google as well.

Now, I bought a domain from Google, and I need to host a simple static website, that will access this Google Function using a Function URL, and show data to client.

It need to be fast and serve many users. ( It is sort of a search engine)

I would like to avoid Firebase Hosting for multiple reasons, but mainly because i want to stay inside the GCP, where i deploy with and monitor everything.

I realized that my options to host this static(?) website with my custom domain in GCP are :

1. Load Balancer - which is expensive over kill solution.
2. Cloud Storage - which (i might be wrong) will be very limiting later if i need to manage paying users. ( or can i just send user ID to the Function using parameters?)
3. Cloud Run - which i am not sure exactly yet what it does.

What is a solution that fit a light web app(html/JS) that can Auth users but connect to a massive Cloud Function using the Cloud Function URL with simple REST?

Also - can i change the URL of that Cloud Function to be my domain without Balancer ? Currently it is like project-348324