Is the use of 'as' in place of 'because' considered to be poor grammar/style? For example
I cannot come with you as I am too busy
I cannot come with you because I am too busy
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In some sentences as can be ambiguous, since it may be unclear whether it expresses time or cause. There is no such risk in your example, however, and you may use either as or because.
As background, causative as seems to be rarer than temporal as, and American speakers and writers seem less inclined to use causative as than British speakers and writers ('The Cambridge Guide to English Usage').
Using causal "as" is not poor grammar. Choosing to use it when it unintentionally creates an ambiguity is poor style (your readers/listeners will eventually figure out what you mean to say, so you've merely wasted a little of their time, but their time isn't as important as your right to express yourself in any manner you choose [irony alert]). If you want to be ambiguous and confuse your readers and listeners, however, then you should by all means be as ambiguous as possible. Scads of native speakers will support that position by saying that "Everybody does it" (mendacious) or "That's what I say, so it must be right" (holier-than-thou) or "What makes you think there's a problem with this?" (dogmatic) or "I don't have a problem with it, so there's no problem with it" (solipsistic). It's a personal preference and, as Barrie England points out, American speakers and writers tend to use it less than British speakers and writers do. That doesn't influence my preference one iota, jot, or tittle. George Eliot (British), Nathanael West (American), Neil Gaiman (British), and Yann Martel (Canadian) are my current favorite writers and users of English.
As Merriam-Webster's Dictionary of English Usage points out, according to Harry Shaw (Errors in English. 2d ed. NY: Barnes & Noble, 1970), "as" is one of the most overworked words in the English language. The very long entry gives all kinds of examples of how the word has been used through the ages. Most of the time, but not always, there's no problem with understanding what it means in context.
I always replace causal "as" (and causal "since") with "because", or else I rewrite the sentence. That's because I believe that there should not be ambiguity in biomedical or other technical writing. It isn't poetry or detective fiction (well, sometimes it's criminal fiction). Clarity, brevity, and plain-style are what I strive for in formal written prose. Ambiguity and other types of unclear writing convey a "Gee, Officer Krupke. Krup you!" attitude to the reader/listener. That's just my personal bias, of course, but no journal copy-editor, editor-in-chief, or reviewer anywhere in the world has ever changed "because" back to "as" or "since" in the thousands of manuscripts that I've revised. For what that's worth.
BTW, in both your example sentences, I'd say "I can't go with you because I'm too busy" and "I can't come with you because our auras don't match". And you're right, there is and has long been an absurd and pedantic debate about using as. Don't bother to get involved. Just choose your poison and drink it.
"As" is a lazy word. It has no backbone. "Because" is much more precise. After all, the useful word "cause" is right there! And for me, "since" always implies not causality but the passage of time ("Since I've lived here..."). The English language is full of words that have overlapping meanings; it's the writer's job to choose the most accurate word in every context.