Is it possible to override the default Gallery Settings form?

There is no way to alter the gallery settings. About the only thing you can do is to override the entire gallery shortcode. You can technically have your users pass any sort of parameters into the shortcode and recognize them with your alternate gallery.

Simply use:

add_filter('post_gallery', 'foo_override_gallery', 1, 2);
function foo_override_gallery($empty, $attr){
    //Extract the attributes
    //Create your own gallery layout
    return $output;
}

By returning any value to the filter, the default gallery is completely overridden. You must return HTML for this to work properly.

I know this isn’t exactly what you’re looking for, but it’s the closest way to customize the output of the gallery shortcode. I dumped the gallery template here sans the override filter that you could put in your new function: http://pastie.org/4293527

Hope this helps.