How to add a HTML menu item to secondary menu with specific/custom hook element

Hi,
I tried adding a looking glass svg icon to my SECONDARY menu triggering the search modal.
The modal is enabled in the customizer, and I verified the data-gpmodal-trigger attribute works when added to arbitrary html elements. Using the Code example from https://docs.generatepress.com/article/navigation-search-modal/ it did
not work to place the icon on the secondary menu.

With the default menu editor, the WP “custom link” allows no custom data attributes and using raw svg as a label looks quite ugly in the menu backend.

So for a clean implementation that can be later customized in the GUI, I created a GP Hook element :

<a class="gp-menu-item ncb-menu-search-icon" 
   data-gpmodal-trigger="gp-search" 
   aria-label="Suche" 
   role="button" 
   style="cursor:pointer" 
   tabindex="0">
  <span class="gb-shape">
    <svg aria-hidden="true" role="img" height="1em" width="1em" viewBox="0 0 512 512" xmlns="http://www.w3.org/2000/svg">
      <path fill="currentColor" d="M505 442.7L405.3 343c28.4-34.9 45.7-79.4 45.7-128C451 96.5 354.5 0 233 0S15 96.5 15 215s96.5 215 215 215c48.6 0 93.1-17.3 128-45.7l99.7 99.7c4.5 4.5 10.6 7 17 7s12.5-2.5 17-7c9.4-9.4 9.4-24.6 0-34zM233 367c-83.9 0-152-68.1-152-152S149.1 63 233 63s152 68.1 152 152-68.1 152-152 152z"/>
    </svg>
  </span>
</a>

The element has a display rule “Entire Site” – it shows up fine if using content hooks like generate_after_content etc.

But I could not get it to show up when assigning navigation hooks. I tried a couple of them, like generate_menu_bar_items or generate_inside_secondary_navigation. The strange thing is: the hook is displayed in the list of active elements in the admin menu bar (see screenshot, upper part).

https://pasteboard.co/2a7xOfROZy7w.png

So the element is shown to be present, but nevertheless there is no rendered html of the element in the page.
Note that the menu is inside a site header block element ( screenshot lower half).

Bottom line – is there a way to get a custom HTML menu item into the secondary nav with GP hooks and elements, or do I have to use custom code in functions.php or a php code element? Here in this forum, mostly code using generate_menubar_items is recommended, but like I said this has no output in my case.

Thanks, best Regards
Ulrich