Can I change "would have been cause" to "would have been caused"? If not, please explain. Does the word cause refer to noun?
This would have been cause to doubt that divine favour has been bestowed on Romans.
You are right: cause is a noun in this sentence, and been is the main verb in this clause.
To use the verbal form of cause, you would have to say something like
This would have caused doubt that divine favour has been bestowed on Romans.
or, less elegantly,
Doubt would have been caused that divine favour has been bestowed on Romans.
The meaning would be somewhat different though: the verbal form suggests confidently that this would have caused doubt, whereas the original sentence merely suggests that a cause for doubt existed, without suggesting that it actually caused any doubt.
No you can not make the change you describe.
[Something] would have been cause to doubt...
Means that [something] would have been the reason for whatever comes next (to doubt that divine favor has been bestowed on Romans). In this case "cause" as you correctly identify is a noun. The [something] in this case is the subject of the sentence (I think so anyhow, but my sentence diagramming is not best).
[Something] would have been caused to doubt...
Would mean something else. Specifically that [something] is being made to wonder "that divine favour has been bestowed on Romans." This actually sounds ungrammatical with "this" or "something" since we generally assume that these pronouns refer to entities which are not capable of "thought" or wonder.