Edit “Locations” field in Search Widget

I was looking into the source code for the public-search-form.php file and I realized that the “Locations” select box uses a wp_dropdow(); function – I am trying to instead use the wp_terms_checklist() to display a checklist of child pages you wish to search through.
I have tried swapping out the functions in the public_Search_form.php file (both vertical and inline) but I keep getting error pages. Can anyone tell me what is required to achieve a checklist dropdown?

Greatly appreciated!