Category child in search bar

Hello,
I’m trying to adapt the search bar with our needs.

Is there a way to select the parent category as default for an specific page? and how to show the child categories in the search bar

for ex.
We have “Médecine Esthétique / Laboratoire / Matériel” Categories (parent/child/sub-child)

We want for the medecine esthetique page, to have in the search bar, “Médecine esthétique” category selected by default (and hide it with css) and both child options to select in dropdown menu as | “please Select \/” | “please select \/” | Enter your search | Search |