What is the most efficient way to retrieve a group of image URL’s from different attachment ID’s of a specific crop size?

I don’t believe there is more efficient way. Your primary object should be keeping things simple while using WP’s API which is exactly what you’re already doing. It’s true some scenarios could become pretty expensive and that is where you could take advantage of WP Object Cache or even Transients API. A good example of cache usage are the default widgets found in wp-includes/default-widgets.php