After fussing several times to get the dynamic menus to work as desired, the ease of hard-coding a single menu won out.
Step 1. A single child theme is now present for the whole network. Next, the menu was inserted. Also, some other key elements were added for the admins of the network.
Step 2. All of the blogs and community forum links will be inserted over time. Some of the major ones are present.
While this took a long time, this stops the issues with subnetwork menus not updating within the network.
I did a little more work on the menu this morning. In particular, some of the forums were added into the menus on the WordPress side.
A little more work on the menu was done today. Specifically, the business discussion areas were linked. Some of the menu numbering was modified. There were also a few errors in missing HTML tags.
There are still many more blogs to link into the common header, as well as navigation for some of the older blogs (to link archives).