Utilise WP_Customize_Image_Control to only show image picker

I suggest you to build your own WP_Customize_Control class that does exactly what you want. You can copy-paste the core WP_Customize_Image_Control code on your own class and modify it to fit your needs.

You can follow Otto’s guide to create a new control.

I personally use the Customizer API to modify the background-position CSS property of the background image using the mouse dragging movement (see the git repo).