This appears to be an elision of "to be" from "such as it might prove to be", which is slightly awkward.
Let's make up a different example to illustrate this.
A proper engineer, such as you might someday prove (to be), always takes into account the economic feasibility of a proposed design. (You're not a proper engineer yet. You might one day prove to be one (or might not). A proper engineer always takes into account the economic feasibility of a proposed design.)
That is to say, the sentence talks about a writer's manuscript, and the remark suggests that the draft work is not yet classified into a form, but will perhaps come to be regarded as a narrative.
Now it's a little hard, at first, to understand the purpose of the remark, because "a narrative" is simply a story consisting of some events presented in some order. It's obvious whether some finished manuscript is a narrative, or something else. It usually isn't something that has to be proven; to call something a narrative isn't to say that it's worthy in some way.
However, it could be that the writer is saying that the manuscript was in fact so badly put together that it cannot be recognized as having the structure of a narrative in its present form: that only after some heavy editing might something which looks like a narrative emerge out of the whole mess (thus "prove to be" a narrative).
This interpretation is supported by the presence of the observation that the writer chose a friend who is biased in his favor to proofread his work, the implication being that a writer who is confident in his abilities would choose an unbiased, honest reviewer.
The remark "the only critic whose judgment he dared depend on" is also quite sarcastic; the writer "dared to depend on" not some sharp, skilled literary critic, but a mere friend who is biased toward giving favorable feedback.
(Imagine a paragraph written about a surgeon, who "dared depend on" nothing other than an old, rusty, dull scalpel.)