Hey Max!
We have a very similar setup. However, for us it is not spread across multiple clients, but across multiple types of client requests (because the same team would work on the request, regardless who the client is).
Would it be possible to make your ‘Asana-sphere’ easier to grasp and track, if you create a team for each client? This way you could sort all of your projects as Asana projects and use the dashboard. At the same time, we implemented a team, which includes everyone in the organization and which holds all the projects that are not linked to a specific client, such as organizational matters and summary projects (Using Asana "summary" projects to view your most important tasks).
Would this make sense for you as well?