Unfortunately, No.
Look, when user clicks ‘Add New’ post, WordPress creates a new post record at the database and then re-open for editing the existing (just new added) post. As you allowed to user to edit posts from selected categories list only, if this new added post will not has any allowed category, WordPress will show “You don’t have permissions to edit this post”. This is a reason why I have to assign automatically to the new added post the 1st item from the allowed list of available (allowed) categories.
So user has to reselect needed category if needed after that.
