The Approach

I think the best approach is to create an image size “on the fly”, just before the images is resized.

You can do that using 'intermediate_image_sizes_advanced' filter hook. That allows you to edit the size to be generated, but being aware of the current image size, that is stored in the array $metadata passed by the filter as second argument.

The Math

First of all let’s write a class that returns the biggest sizes for a specific ratio.

class ImageRatio {

  private $ratio;

  function __construct($ratioW = 4, $ratioH = 3) {
    $this->ratio = array($ratioW, $ratioH);
  }

  function getLargestSize($imgW, $imgH) {
    $inverse = false;
    // let's try to keep width and calculate new height  
    $newSize = round(($this->ratio[1] * $imgW) / $this->ratio[0]);
    if ($newSize > $imgH) {
       $inverse = true;
       // if the calculated height is bigger than actual size
       // let's keep current height and calculate new width
       $newSize = round(($this->ratio[0] * $imgH) / $this->ratio[1]);
    }

    return $inverse ? array( $newSize, $imgH ) : array( $imgW, $newSize );
  }

}

Class usage

The usage of the class is pretty easy:

$ratio = new ImageRatio(4, 3)

$ratio->getLargestSize(1000, 500); // return: array(667, 500)
$ratio->getLargestSize(1000, 800); // return: array(1000, 750)

In Action

At this point, we can make use of the class to calculate on the fly an new images size, based on the image is being uploaded

add_filter( 'intermediate_image_sizes_advanced', function( $sizes, $metadata ) {

   if (! empty( $metadata['width'] ) && ! empty( $metadata['height'] ) ) {
      // calculate the max width and height for the ratio
      $ratio = new ImageRatio( 4, 3 );
      list($width, $height) = $ratio->getLargestSize( 
         $metadata['width'],
         $metadata['height']
      );
      // let's add our custom size
      $sizes['biggest-4-3'] = array(
        'width'  => $width,
        'height' => $height,
        'crop'   => true
      );
   }

   return $sizes;

}, 10, 2 );

Using the New Size

$image = wp_get_attachment_image( $attachment_id, 'biggest-4-3' );

Note

Of course, this works for all the images that you upload after the code is in place. For older images, you should re-generate the thumbnails, on the fly when they are used, or in bulk using one of the plugins available on the web.