But there's nothing written about the imperial edition anywhere.
I'll check on that.
So is there no way to get the Imperial Edition from GOG via KS backing?
There was a mistake with the GOG codes for some D:OS backers; the Larian Vault download was a quick fix while they sorted that out.
You can contact GOG support to get it on your GOG account. From the GOG DC release topic:
Anyone who backed the Original Sin Kickstarter and claimed his or her copy of Divinity: Dragon Commander on GOG and *didn't* get the Imperial Edition code issued, please contact Support and we will give you a free code for the DLC. In the subject of your ticket, please put, "Imperial Edition Upgrade for Kickstarter Backer".
Larian is working on a fix for the save compatibility issue with the GOG version.