Thanks for the links! There are several ways to accomplish this, and some require more coding than others. Let’s look at the simplest route first, which involves no code editing, and see if that method might work for you.
In the Customizer’s Portfolio panel, there’s a section where you can add text, including HTML, that will be displayed on all portfolio archive pages.
You could add your portfolio types menu there, it would look something like this:

Example:
<a href="https://tflanagan.us/shareyourpoem/project-type/middle-school/">Middle School</a>
I’ve added just one link to keep the screenshot simple, but you could add as many as you like, following the example above.
What this does is add the text in that box to all the automatically generated portfolio archive pages, including the main portfolio archive page here: https://tflanagan.us/shareyourpoem/portfolio/ as well as all project type and project tag archives.
To use this method, you’d also need to change the link in your menu that currently goes to https://tflanagan.us/shareyourpoem/poems-2/ to the automatically generated portfolio archive instead: https://tflanagan.us/shareyourpoem/portfolio/
Let me know how it goes!
Here’s what it looks like on my test site:

You could always change the look and feel of the links with custom CSS as well. If you need any further help with this, just let me know.