W B Thank you for your feedback.
That's right, there is no "When" trigger for label added, but you can add a label as a condition which will allow you to automate based on labels.
For example, choose the when "Conversations is closed", add a condition for the Invoice label, and the action to "forward to email" (perhaps? I'm not 100% sure what you meant by "send directly to our online admin tool"). With this rule, you would simply need to add the label and close the conversation for the rule to trigger. Correct me if i'm wrong but this would basically be the same as what you wanted but in another order.