I doubt that resized images will be progressive jpegs.
/var/images directory is used to cache resized images. For example, if X-Cart knows that the active theme displays product images at X*Y, A*B and M*N dimensions, it will resize every uploaded image to these dimensions and store them for future use. Sometimes this happens not at the time when an image is uploaded, but at the time it is displayed in that particular width and height the first time.
Do you have ImageMagick installed on the server? I'm not sure if there is a solution for the GD library for PHP, but I think it can be done for ImageMagick (with a minor tweak) as is described
here. So, you are to do the following:
1. Install ImageMagick on the server
2. Edit etc/config.php and specify your image_magick_path
3. From a custom module decorate the "execResize()" method in \XLite\Core\ImageOperator\ImageMagic class so that it calls the imagemagick binary with the extra "-interlace Plane" parameter.
Does this help?