All three examples: scan, scanned copy or copy could legitimately be used in the situation you describe (listed in my likely order of preference, assuming nothing else to influence my decision). Note: if I was sending a physical letter, with a photocopy or similar of a document, then I would favour "copy".
From the OED:
3. An image, diagram, etc., obtained by scanning
Source: OED, "scan"
II. A transcript or reproduction of an original.
2. A writing transcribed from, and reproducing the contents of, another; a transcript.
3. A picture, or other work of art, reproducing the features of another.
Source: OED, "copy"
(The OED doesn't directly acknowledge the existence of electronic scanners or photocopiers in either entry, but the sense should be clear).
My (slight) preference for "scan" over "copy" is probably as much due to my age as etymology; coming as I do from a time when "copy" almost exclusively meant physical copy, and only later did such things as electronic copies exist. With that history, "scan" emphasizes the electronic, somewhat intangible nature of an attachment.
For people who have probably never seen an actual, dedicated photocopier this distinction between physical copy and electronic scan is likely to diminish and, perhaps, disappear. (For them, making a (photo)copy means printing the file that has been scanned!)
For me, 'scan' seems accurate as it's a noun, and says exactly what it is. But 'scan' also feels like it refers to the instance of the action of scanning...
You are quite correct: scan can be both a noun (the thing produced by the act of scanning) and a verb (the act of scanning, that produces a scan). Exactly the same applies to "copy". However, that the same word can be used as a verb does not invalidate it when it is being used (correctly) as a noun.
As far as "scanned copy" is concerned: I personally don't feel it is "too much for a quick email" (but, again, that is probably partly due to my age and a tendency for more formality, even in emails). However, I'd probably only use it if I'm directly contrasting a scanned attachment with a separate physical copy. For example, there are occasions where you may need to send a physical copy of a document by post, but can expedite processing by emailing an electronic copy. In such a case, I might well say:
Please find attached a scanned copy of Form XYZZY; a copy of the original will follow by post.