I tried using the web service, but it complained that it needed a login/pw from the user that is assigned the task. A username is no problem, but no way can I get a password (that I am aware of, anyway). It complained about this despite the security tab of the ProcessManagementTaskService having 'allow anonymous access' turned on and still complained if I turned off the need to authenticate.
This caused me to look for yet another solution and I found one using what I am guessing is invoking the exact same thing but in a different manner. I dragged the "completeTask" operation of the "Task Notification 1.0" service from the Process Management category. I needed to supply the task ID and the username of the assigned user. This seems to have done the trick.

Thanks for pointing me in the right direction!