Hi,
In general custom works like this are beyond our support, but I can suggest how to start with this problem. In general you have to create in the module duplicate of the menu which will be generated as the select dropdown list. Then you have hide this dropdown in the desktops and show on the tablets and mobiles (tablet.css/mobile.css) where the standard menu will be displayed. The last thing to do is create some background image for the button (that's why you should wrap the select dropdown list with some container) and hide the dropdown list using opacity set to 0 - then the list won't be visible but after clicking on the background it will display.