This is something you can achieve with the Advanced Custom Fields plugin. The trick is to use the ACF_LITE mode, which would let you define a dynamic list of user profile fields from functions.php. You could iterate over each post type and display a checkbox for it. Then use the edit_user_profile_update hook to call $user->add_cap('edit_[resource]') or $user->remove_cap('edit_[resource]'), etc.
