Do you want to move the settings and the search icon away from the search input for mobile devices?
I’m afrait that is not possible like that without major customizations, as the fields are part of the form in a flex container.
You should try increasing the container width of the box, so the input can get wider: https://i.imgur.com/e859fpr.png
So it can get to the edge of the screen. Decreasing the padding on that could also help.
For the plugin you can set a fixed width (either pixes, %, vw units) here too: https://i.imgur.com/isa6jFX.png
By default the search input occupies 100% width of the container, but sometimes it might be conveniet to set a fixed width.