Hi there,
you can use a Block Element – Hook:
https://docs.generatepress.com/article/block-element-hook/
1. In Appearance > GeneratePress -> Activate the Elements Module
2. In Appearance > Elements -> Add New –> Block.
3. In the sidebar settings set the Hook to: menu_bar_items
4. In the Display Rules set the Location to where you want it displayed eg. Entire Site
5. Then add your button block to the editor and style it how you want etc.
NOTE: This will also appear in the mobile navigation header.