Use an American key from Amazon in the Origin client which is identified as British

It works one way, but not the other. You can use the keys gained in Steam (Right click on game in left-hand menu, View Game CD Key) and activate them for your EA Games account by registering them at https://activate.ea.com, which makes them then appear in your Origin games list for installation. (This applies to most - but not all - EA Games that are available on Origin)

You cannot use Origin codes to apply games to Steam, and Steam won't let you buy DLC for games you don't own though Steam.

[2015 update: You now activate them on Origin though the client. Origin menu, Redeem Product Code.]

In many cases for EA games you can use the key from Steam to also get the game on Origin. This is an automated process, meaning it is quick and can be done at any time, but doesn't work for all games. If this doesn't work you can consider using Origin support and hoping for the best. You can also just buy the DLC and hope for the best, it may still work. But actually getting the game you bought on Steam to show up as owned on Origin is the safest way to ensure that the DLC from Origin will work.

But first – how to acquire a CD key from Steam? If the game supports it, a “View CD key” option will be found when right-clicking it. In the example below, we can see that Crysis 2: Maximum Edition is one such title that does.

[...]

Here is a list of EA games / DLC and their confirmed status:

 Game Title                             Success Notes 
 Alice: Madness Returns                 Yes     Success 
 Battlefield: Bad Company 2             Yes     Success 
 Battlefield: Bad Company 2 – Vietnam   Yes     Success 
 Battlefield: Bad Company 2 – SPECACT   Yes     Success 
 Bulletstorm                            Yes     Success 
 Burnout Paradise: The Ultimate Box     Yes     Success 
 Command & Conquer: Red Alert 3         Yes     Success 
 Command & Conquer 4: Tiberian Twilight Yes     Success 
 Crysis                                 No      Origin Rejects Key 
 Crysis Warhead                         Yes     Register on Website* 
 Crysis 2                               Yes     Success 
 Crysis 2: Maximum Edition              Yes     Success 
 Darkspore                              Yes     Success 
 Dead Space                             Yes     Success 
 Dead Space 2                           Yes     Success 
 Dragon Age: Origins                    Yes     Success 
 Dragon Age: Origins – Awakening        Yes     Success 
 FIFA Manager 09                        No      Steam Doesn’t Provide Key 
 FIFA Manager 10                        No      Origin Rejects Key 
 FIFA Manager 11                        Yes     Success 
 Gatling Gears                          Yes     Success 
 Mass Effect                            No      Origin Rejects Key 
 Mass Effect 2                          Yes     Success 
 Medal of Honor                         Yes     Success 
 Mirror’s Edge                          No      Steam Doesn’t Provide Key 
 Need for Speed: Hot Pursuit            Yes     Success 
 Need for Speed: SHIFT                  No      Steam Doesn’t Provide Key 
 Need for Speed: Undercover             No      Steam Doesn’t Provide Key 
 SHIFT 2: Unleashed                     Yes     Success 
 Spore                                  Yes     Success 
 Spore: Creepy & Cute                   No       Steam Doesn’t Provide Key 
 The Sims 3                             Yes     Register on Website* 
 The Sims 3: Ambitions                  Yes     Register on Website* 
 The Sims 3: Fast Lane                  Yes     Register on Website* 
 The Sims 3: Generations                Yes     Register on Website* 
 The Sims 3: High-End Loft Stuff        Yes     Register on Website* 
 The Sims 3: Into the Future            Yes     Success 
 The Sims 3: Late Night                 Yes     Success 
 The Sims 3: Outdoor Living Stuff       Yes     Register on Website* 
 The Sims 3: World Adventure            Yes     Register on Website* 
 The Sims: Medieval                     Yes     Success 
 The Sims: Medieval – Pirates & Nobles  Yes     Success 
 WARP                                   Yes     Success