Hi Leo,
I am sorry for not looking at it earlier.

Today, I looked at it.
Here is what I did.
1. I registered a custom post type article like this

function buddydev_register_post_types() {
	
	register_post_type( 'articles', array(
		'labels' => array(
			'name'			=> 'Articles',
			'singular_name' => 'Article',
			
		),
		'public'	=> true,
		'slug'		=> 'articles',
		'supports'	=> array('title', 'editor', 'thumbnail', 'author'),
	) );
}
add_action( 'init', 'buddydev_register_post_types' );

Then I went to BuddyBlog settings and chose this post type as selected.

After that I tried to upload with Subscriber User account and It worked perfectly.

Note:- I am using latest version of BPSFEP, BuddyBlog, BUddyPress, WordPress and testing it on non multisite.

From the above test, I believe you are using some 3rd party plugin that has been conflicting with permissions and causing it.

Can you please check by disabling a few or replicating on a test server?

Thank you
Brajesh