How do you render_callback for register_block_type to a method in another class?

Option 1 is the solution, here’s a smaller example:

$obj = new Obj();
....
register_block_type(
    'my-plugin/my-block',
    [
        'editor_script' => 'editor-script-handle',
        'render_callback' => [ $obj, 'block_save_function' ]
    ]
);

In that code [ $obj, 'block_save_function' ] is equivalent to $obj->block_save_function(....

The important thing to note, is that render_callback‘s value is a PHP callable type. You need a reference to the object, and the name of the method to call. This is true of all callbacks, such as when you add an action or filter.

Likewise, option 2 and 3 can be fixed by using the same syntax.

To learn more about the different ways to create callables, see the PHP official documentation here https://www.php.net/manual/en/language.types.callable.php