Ok, it appears to be working. I changed your loop to look like this:
//Fetch the gallery for the selected post
$gallery = get_field('gallery', $post_id);
if($gallery){
$i = 0;
foreach($gallery as $image){ //loop through the images in the gallery
$gallery[$i]['constrain'] = get_field('constrain', $gallery[$i]['id']);
$i++;
}
}
Thanks for all of the help!