What export size should I use from Lightroom for the best Instagram image quality?

I think the 612 pixel dimension is some way out of date. This post from Instagram's blog is an old post introducing a new feature since version 2 (we're on version 6 on iOS now), and it says that the resolution has increased from 640px square to 1936px square on the iPhone 4, so it looks like even around 2-3 years ago, the dimensions of Instagram images was increased to 1936px square.

It looks like you found the 612px dimension from a google result that came from Instructables. Even that article is approaching a year out of date because the example given by the author links through to a flickr photo that was uploaded in August 2014, but I think they were using wrong information in the first place.

I think that instagram isn't comrpessing your pictures, but is actually enlarging them to somewhere around 4 times their previous width and 4 times their previous height, therefore actually about 16 times bigger in area.

I would probably try not to limit the file size too much when exporting from Lightroom, unless you test the exported results first.

Try cropping your image to 1936px square first, then export using these dimensions