Yes! This is possible but it will require custom coding and help from your developers.
To do this, Enable the 'Clickable Trigger', and switch it to 'Custom Mode'.
From there, you can choose an HTML ID or Class selector (your web developer will know how to add this in your Shopify theme code) that will trigger your Widget when a specific menu item is clicked
Note: Sample mobile menu sidebar where your popup button could be added.
What does a mobile sidebar look like?
How can I customize my mobile menu sidebar?