It seems that Instagram posts can now be seamlessly embedded into a WordPress post without any display issues on various devices. If you haven't already, check out this link for more information:
Additionally, if you use Embedresponsively to embed an Instagram Post on a WP post/page, you have the option to customize the styles and decrease the padding-bottom.
.embed-container {
position: relative;
padding-bottom: 120%;
height: 0;
overflow: hidden;}
.embed-container iframe, .embed-container object, .embed-container embed {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%; }
<div class='embed-container'><iframe src='//instagram.com/p/dnQi4EGuZx/embed/' frameborder='0' scrolling='no' allowtransparency='true'></iframe></div>
The default code provided by Embedresponsively is as follows:
<style>.embed-container {position: relative; padding-bottom: 120%; height: 0; overflow: hidden;} .embed-container iframe, .embed-container object, .embed-container embed { position: absolute; top: 0; left: 0; width: 100%; height: 95%; }</style><div class='embed-container'><iframe src='//instagram.com/p/dnQi4EGuZx/embed/' frameborder='0' scrolling='no' allowtransparency='true'></iframe></div>