Photoshop CC Animated GIF pixelated by Instagram

How many frames does the animation have, and what is the file size after exporting?

I'm thinking the pixel size you mention is on the high size, and likely to lead to large file sizes begging to be compressed.   With 30 frames, I try to keep the pixel size below 600 pixels on the long side.  Depending on how complex the images are, and the number of colours you need to maintain, this will still lead to file sizes above 1Mb, and this is just one component of the web page your GIF will be displayed on. 

If you need to keep all 256 colours, a 1080 x 1080 pixel animation with a bunch of frames can easily be >3Mb.  If you can control the file size before uploading to Instagram, then you might keep it under whatever size triggers their compression algorithm, which by the sound of it, is not kind to your artwork.

How about uploading the animation, as exported from Photoshop, to this thread?

Specifically for video problem, ive found this suggestion:

"It is happening because Instagram has it's own converter and it makes your video's smaller in size, so that it will save bandwidth and load faster.

Try optimizing your video before you upload it to Instagram. i.e: Export in .MP4 using h.264, at a bit-rate that will make your file size be less than 15MB. This is what i use most of the time and it works for me. Try playing with you export settings and see what works best.

But for most online videos .MP4 with h.264 encoding works best."

Please, try a resolution of 1080 x 1080 @72 sRGB

Then Experiment with different quality settings between 75 & 100 as Instagram will apply standard compression.

In alternative there is a lightroom plugin to export directly to Instagram:

Instagram Plug in