Brandon,
CSS media queries will not work in this situation as the widgets code sets the width programmaticly based on the controller state (minimized or maximized). When I tested in the WAB Preview on mobile devices it worked fine for me. If you only want the width adjusted on mobile the you can add some code to check for the WAB specific window.appInfo.isRunInMobile property.
I tested using Chrome developer tools and using mobile emulator and it worked great using this code changes to the sidebarController widget.js (added lines 2 - 6):
getWidth: function(){
if (window.appInfo.isRunInMobile) {
this.maxWidth = 200;
} else {
this.maxWidth = 365;
}
var box = html.getContentBox(window.jimuConfig.layoutId);
var w;
if(box.w * 0.8 > this.maxWidth){
w = this.maxWidth;
}else{
w = box.w * 0.8;
}
return w;
},