I am trying to write something along the lines of 'the Daibutsu is made out of bronze.' I am using です・ます体. In this case, which word would be more appropriate to use? Is there even a difference?
First, while English speakers distinguish "copper" and "bronze" rather strictly, Japanese people do not use 青銅 that often. "Bronze medal" is just 銅メダル, and "bronze statue" is usually just 銅像. If you ask a native speaker "What is a 10-yen coin made of?", the response is probably 銅, not 青銅 nor ブロンズ.
Typically, ブロンズ(像) is used for Western-style bronze statues, and it's rare to use ブロンズ to refer to Buddhist statues. The following image search results support this:
Therefore, even though 青銅 and ブロンズ are chemically the same, it is much safer to use 青銅 or just 銅 when you are talking about daibutsu. However, this is not a strict rule. For example, in the Wikipedia article for 牛久大仏, ブロンズ is used many times for a giant daibutsu, and this is probably because international comparison was important when applying to the Guinness Book.