Not sure if that was suggested or even considered before, but...
How about using those pesky torrent sites full of so called pirates to your advantage? Those are really good at spreading the word out, but it would require taking some risk. Don't know how difficult would it be to prepare a special edition of the game solely for that purpose but my ideas would be for example:
- Spreading out the demo to as many torrent sites as possible, but with longer time limit - say 10 hours. Why? 3 hours, as it is now, is rather short for AI War, you can barely scratch the surface, especially if you need to go through tutorial first. Actually, tutorial could be without a time limit itself. Anyway, if someone doesn't like the game in 10 hour mark, chances are he won't buy it or even tell anyone about it anyway. If one does however, it might be better encouragement to buy full version once he committed two or three evenings on it already. 10 hours is not really long enough to finish a single game for the first time, could be one more prod into getting the real deal.
- Creating a stripped down version with only basic options enabled but without a time limit at the same time. So only simple ships selection, two or three AI types per difficulty level, one maps layout, limited factions and modifiers, no achievements etc. All those options that are taken out should still be visible and described properly, so that you know what are you missing from the game. Having game in such state out on torrent sites could generate some attention.
In addition to releasing special demo/stripped version to torrent sites, you could try advertising your move via press releases (including TorrentFreak for example), stressing out that you're giving away large portion of the game for FREE (like: "get 10 hours of game time for FREE" or something like that) - don't call it a demo, make it feel special and out of the ordinary. You might also think about a discount code hidden (not too deep so people can find it) in the torrent release - word of mouth might spread thanks to it.
Added bonus of that approach is that if it catches on, the torrent will live for itself without much need to provide bandwidth for downloads and taking care of the attention around it.
There is of course a matter of the game getting cracked, but those not willing to buy it won't buy it anyway, while those who get the cracked game and like it enough might buy it eventually and spread the word some more in the mean time.
Just a quick ideas that I'm sure need refining and considering the potential risks, but I do think it is worth exploring in one way or another.