BuddyPress registration page is quite big, it has a lot of inputs.
You can install a widget, called (BuddyPress) Log in. Here is info about it.
If you use Jetpack, there is a module there, that gives ability to define widgets visibility. Here are other plugins: one, two & etc.
Thus you will have ability to define, that this widget should be displayed only on Registration page in sidebar (or any widget zone that is defined in your theme).
Or you can manually edit BuddyPress template files and include a shortcode or PHP code to display a login form in any place of your registration page. See docs for that.
If you know what you’re doing inside register.php it can be done. Why would you want to go to this trouble when yoursite.com/wp-login.php and yoursite.com/register.php work just fine?
