Hi, this might help for people trying to do what you want to, or something similar. It certainly seems to fulfill my needs, which I think are the same as yours.
All I did was add a Template Tag in the header file.
I added:
<li class="page_item"><?php list_cats(); ?></li>
just before the last </ul>
tag, at the end of the header file.
In the list_cats documentation, there’s the arguments to customize it.
I hope that helps anybody looking for this thing. I also hope I won’t suddenly find that this doesn’t actually work and have to spend 4 more hours trying to find out why.