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 DEMO
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