theres no simple way to move the Navigation Search inside the off canvas panel.
Instead you would add a Search Widget to the Off Canvas Panel Widget Area:
I have learnt something with it. It didnt work for me in the menu – looked weird with my custom button which I hooked in via the block element feature.
At the end I have used Leos code and adjusted it with -3.4em as the icon on mobile was kind of in the middle, now it looks better aside the menu icon.