I eventually figured out a workaround.
The core issue is that the height of the header will prevent clickthrough even if there are no objects in the header that are in the way.
So I made the header shorter. The longer version of the header has its background split into two strips, the second of which is popped down into place on page load. The longer header is basically hanging outside of the header’s bounds. When switching to the smaller header after scrolling down, there’s no longer a dead zone because the header’s box is the smaller version of the header’s vertical size.
To accommodate this, I’ve added a small spacer at the top of each page so the page contents won’t overlap with the longer version of the header. This spacer is the height of the old dead zone. (59 pixels on Desktop & 89 pixels on Mobile)