Hi,
This isn’t possible natively — the Site Header block doesn’t have a separate menu selector for desktop and mobile.
As a workaround, you can add two Navigation blocks inside your Site Header, assign a different menu to each, and then use the block’s visibility settings to control which one displays at which breakpoint. The mobile nav would live inside the overlay panel, and the desktop nav would be hidden at mobile breakpoints.
It does require a bit of manual setup, but it gets the job done. Let us know if you’d like guidance on the specifics.