And I was making it way more difficult than it needed to be. Use an FTP program to access the style.css file and simply add the !important rule. Voilà, it works!
#bgmp_map-canvas
{
width: 100% !important; /* The width and height here are just defaults in case the JavaScript can't load. They're overridden with the ones on the Settings page when the map is created. (Good ole !important saves the day again.) */
height: 400px;
}