I'm not sure I'm following, so I'll answer the question for both of the ways I can interpret that question.

If you mean to use site: operator in your , I don't think you can.

You can use it to choose your instance once, but why not to always use it?

You can use it in the URL bar. Except the domain itself, the rest of the URL is remaining the same.

https://gооglе.соm/search?q=nyan%20cat would become https://farside.link/whoogle/search?q=nyan%20cat.

(You can do https://search.garudalinux.org/search?q=nyan%20cat, but Farside is fantastic.)