I am willing to throw money at this problem, because right now my nightclub has no promotion on Instagram, and sadly, that matters. But I would like a cheaper solution than "hire a full-time staffer to re-post everything by hand".
I have emailed every company I could find who offer a post-to-Instagram service (Later, Hootsuite, Onlypult, Crowdfireapp, Iconosquare, Bufferapp, Skedsocial, Sproutsocial) and asked them:
Does your service provide the ability to post to my Instagram Business account via an API? That is: I want my server to contact your site and say "post this image or video to my Instagram right now", without me needing to use a GUI or web app to manually schedule it.
Every one of them said "no". Though most or all of these companies appear to have access to the sekrit Facebook API, all of them require you to manually schedule each post by hand in their custom, idiosyncratic online calendar.