Instagram has one of the least interesting APIs of all popular services, with basically just 2 GET endpoints or something. So if you do want to build this, you're gonna have to do a lot of reverse-engineering of the smartphone application. Probably a fun project on just it's own.
Guessing they are redirecting to instagram.com based on user-agent or something, because I end up on the lame read-only desktop website. Does m.instagram.com allow you to post content and everything the mobile app allows you to do?
Sorry for the late answer. I can post content from m.instagram.com if I change the user-agent as you mention, but this is quite easy to do with the browser default dev-tools, just press F12 and change the device to any mobile. I think the only thing it does not work is the stories.