Like many others, I only got the two DLC codes from Steam, redeemed them in the Origins launcher and couldn't connect online because I needed the base game on both Origin and Steam to play the Steam version online. Was in contact with EA Support (live chat) and he actually activated the base game for me on Origin. Took some time but works now.
I recommend people to contact the live chat, and be very clear about that you know that the base game for Origin IS supposed to be included in the Steam purchase. In fact, I had to repeat that multiple times before he finally activated my product on Origin.
