Capability for overwriting/uploading files

I can’t figure out a capability configuration.

For admins and super admins (network) overwriting files etc. works fine.
However, I also created roles for users that are only allowed to manage downloads, nothing more.
For this to work I modified the capability_type parameter for the wpdmpro post type.
(Code example below)

add_filter( 'register_post_type_args', 'register_wpdmpro_args', 10, 2 );
function register_wpdmpro_args( $args, $name ) {
	if ( 'wpdmpro' === $name ) {
		// Custom capability for downloads.
		$args['capability_type'] = $name;
	}
	return $args;
}

https://codex.www.ads-software.com/Function_Reference/register_post_type#capability_type
This way I can separate edit_post and edit_wpdmpro capabilities.
Unfortunately users who can’t access posts cannot overwrite wpdmpro posts.
Is there a better way to configure such roles/capabilities?

Recap: The idea is to create a role that ONLY has access to the downloads, nothing more.

Thanks in advance!
Cheers, Jory