So I need to download the header.php file from the parent theme using FTP and then just transferring it to my child theme?

Yes

Would you recommend this?

Yes

Will this affect a lot for future parent theme updates?

Not until developer makes major changes in the parent theme header.php file in that case you have to also update your child theme header.php file.

The only way I have found is by editing header.php, any recommendations?

Have you tried following solutions?

https://wordpress.org/plugins/google-tag-manager/

http://www.wpbeginner.com/beginners-guide/how-to-install-and-setup-google-tag-manager-in-wordpress/