I think if we were to interpret the modified phrase in your sentence, you would get something like this:
Falling down bloodied.
Which, in my opinion, doesn't make much sense here. By turning 血まみれ into an adverb, you are modifying the verb, as per its name. "Covered in blood" isn't really describing how 山村さん fell, it describes 山村さん themself. The original sentence allows 血まみれ to become a "state of being" by adding になっている to it.
If you were to break down the sentence into separate statements, this is likely what you would get:
Yamamura-san, who was covered in blood.
Yamamura-san, who was collapsed.
倒れている and 血まみれになっている have the same functions here. They describe the state that 山村さん is in.