Programmatically install and activate child-theme

Yes, you can activate the child-theme dynamically. To do that follow the steps

  1. Move the unzipped child-theme to the themes folder
  2. Find the name of the child theme(I guess you already have that)
  3. Then Update the options table by update_option('stylesheet', 'NAME OF THE CHILD THEME')

NOTE: You need to be sure that the active theme and child theme is compatible with each other.