Is it possible to tweak list view search box (Modern) and add additional filter using SPFx?
There are no supported ways to tweak default list view search box using SPFx.
If you want a permanent filter on list view, you can apply it from View settings.
Documentation: Use filtering to modify a SharePoint view
If you want a temporary filter/search via textbox, you can do this using query parameter in URL like:
Note: It will open a SharePoint list in search mode.