Owl Carousel does not respect Image Styles as set in node display in custom node file
Just to summarize my comments:
 1. after getting the images' URLs, you can run them by image_style_url() function to get the URLs to the styled images. 
 2. you need to update the foreach() function to include an index such as foreach($image as $key=&gt;$value), and use the $key index to loop through the image array.