@dgw has the right idea to remove the export buttons, but I wasn't happy with the "and I'd like to add a new one" suggestions until I realized I should just make the buttons outside the graph. Unless your data is static, you don't really know if there's room to display your controls.
The best way to replace the hamburger icon is to disable the navigation buttonOptions, then create your own menu and customize the context one by one as stated in the documentation. From here you can use any icon you want with your own dropdown menu.
This disables the hamburger icon.
navigation: {
buttonOptions: {
enabled: false
}
}
This is how you customize export options for your own list.
The best way to do this is to update the exporting.buttons.contextButton.menuItems array to only include the menu items you want. Below is an example that excludes the "Print Chart" and "View Full Screen" options.