Yes, I would rather use my search site domain, and no subdomain, too. But it doesn’t have to be a big deal, particularly if you use a separate WP installation in multisite for your search.
search.petraforu.com
homefinder.petraforu.com
newnest.petraforu.com
Etc.
even use www. for the subdomain www. petraforu.com
Redirect petraforu.com to the chosen subdomain.
It is a little sloppy, but brings the Platinum benefits, which are many.