Hi there,
you could use a Hook Element to add the Button inside the Navigation.
https://docs.generatepress.com/article/hooks-element-overview/
At this time you would need to create 2 hooks.
1. inside_navigation
for desktop.
2. inside_mobile_header
for mobile.
It will require some CSS to position it. Happy to help with that.
In the next GPP 1.11 update we have added a hook that displays on both nav and mobile header and removes the need for CSS positioning. We look to release GPP 1.11 for Alpha testing in the next 24 hrs.
The HTML for the modal could also be hooked in after_footer
for example.