Hi,
Thank you for the details, it helps a lot!
The issue seems to be the actual widget container for the search bar. In the actual front-end source I am seeing additional HTML tags injected into the search bar (paragraphs and line breaks), which cause the “padding” of the input field. It happens if a text based widget is used for the plugin container, as it treats the plugin code as text, and automatically appends the paragraphs.
To solve this, change the Visual Composer widget from Text to either HTML or a shortcode widget and place the plugin shortcode there – it will resolve the problem.