No they are words with quite different meanings. "Against" is the correct word to use here. If you're "amid" something it means your surrounded by it or you're in the middle of it. A backdrop is a stage prop, often a painted curtain, used in theatre at the back of the stage to place the actors into a certain environment such as a forest or a living room. The word is often used to describe the environment people find themselves in or can be seen in the distance. So, even though you can be surrounded by the environment and the environment can be referred to as a backdrop, because that word is related to the stage you cannot be surrounded by it or "amid" it.