It seems like it .. i did a quick test also and the Tabs dont load until you select them from the sidebar.
If you add the code to the <head> then no access to scope.
Interesting finding you did about the inner workings of the Dashboard but without a way to manipulate that code i think we are stack to adding a hidden template to each Tab .. and also add code to check if your html elements are already injected to the top navbar.
Similar Thread here:
Or an alternative is to look into node-red-contrib-uibuilder
that gives you the flexibility to create your own dashboard from scratch with full access to your own code
but that would require some knowledge of html, css, javascript or any front-end language
