Re: Child theme + custom header + functions.php
https://codex.buddypress.org/theme-development/building-a-buddypress-child-theme/ says:
“If you create a blank functions.php file in your child theme, the parent theme functions.php will still be loaded. This will allow you to inherit existing functions, but also add your own. You must make sure you give your child theme functions a unique name, otherwise they will clash with the parent.”
In other words, you CAN create a functions.php file in your child-theme, but the functions IN that file must have unique names.