So, I have the base game but can't enter the giveaway because SteamGifts says I don't. I'm clueless here.

Yeah, you can't give this away on SG. There's an issue with how Steam is listing the game. I have two keys for this and can't GA them here. It's because originally the base game was for-pay and then it went free. For whatever reason, even if you have the free game installed, Steam doesn't see the pack as being associated with it...I think it's still looking for the old for-pay game, which may have a different id. Dunno. In any case, this prevents anyone from entering. ლ(ಠ益ಠლ)