Hi there,
all Instagram plugins will make 3rd party requests so they create some performance issues. Of the ones i have used, this came up tops:
https://en-gb.wordpress.org/plugins/instagram-feed/
The developer is active as well which is nice. Some features do require the premium version.
For image sizes Right Click over the image and Inspect. In the Developers console it will highlight the element, whilst hovering over it will display the image size being displayed. Look for the largest image being used. In my quick look around the site the largest image size was 693px being used in your post archives.