Sorry for the long delay following up. Summer happened.
I didn't try doing session management via key, since the config files are full of warnings not to do it.
We decided to stick with the old method of using a URL to configure each tool. If a later version of Moodle stops supporting this, I will have to hope IT is willing to configure the load balancer to work. (I suspect they will have to, as WeBWorK is not the only LTI tool we use.)