“Would we be” would be incorrect, as, even in spite of carelessness and cowardice, we actually are on the brink.
I would agree that the two parts of that phrase are somewhat disjointed, and should be interpreted as being somewhat poetic. The second part, to me, really represents a kind of wistful exclamation, as though it could also be written as:
yet with how many things are we upon the brink of becoming acquainted - if only cowardice or carelessness did not restrain our enquiries!
Or to inject the phrasing you proposed, we could say:
yet with how many things are we upon the brink of becoming acquainted, and would (we) be already acquainted if cowardice and carelessness did not restrain our enquiries.
Cowardice and carelessness prevent the acquaintance from being made, but do not prevent us from being on the brink.
I’m going to propose another way to understand this.
yet with how many things are we upon the brink of becoming acquainted
This statement, although not ended with a question mark, is a question. It’s exclamatory and rhetorical; the second part of Shelley’s statement about cowardice and carelessness make it clear that the answer is known and should be assumed - there are indeed many things with which we are on the brink of becoming acquainted.
We could therefore replace the question with an answer, or a statement of fact, such as:
yet there are many things with which we could become acquainted
I have interpreted saying we are “on the brink” as meaning that we could make these acquaintances, perhaps quite easily; we are close.
If we append to this the other part of the sentence, we see that it makes sense:
yet there are many things with which we could become acquainted if cowardice or carelessness did not restrain our enquiries
I’ve also removed the comma, as in this more literal version it is no longer necessary.