I believe the issue is caused by the coding being shared between the desktop and mobile variant, similar to what happens with form scripting. In fact when I tried this on a simple two variant site the desktop showed the widget twice on one variant and not at all on the other!

A possible workaround would be to create an independant web page containing the script and call up this web page in an iframe. I think this might work as the form opens a new window once the car model is selected but I'm slightly unsure re the size of the car model drop-down menu.

<iframe src="http://parkeston.com/groove-car/index.htm" frameborder=0 width=340 height=274>
<p>Your browser does not support iframes.</p>
</iframe>
IFRAME DEMO