While by no means a biblical scholar (or any sort of scholar, for that matter), I'd tend to agree with riley that the verse describes attributes rather than the description of a specific bow, given the context (feet of a deer, etc.).
That said, after working around brass and bronze for 30+ years, any man who could pick up and lug around a bow of bronze would be a
man's man in my book. I'm not sure you could make a "functional" bow in bronze that wouldn't require three men and a boy just to move it around. Steel, on the other hand... India developed and used steel bows, and by some accounts the were both beautiful and functional. A couple of links, there are more, but my sieve memory is extra leaky right now... Also, these would be out of period.
.atarn.org/india/maharaja.htm
margo.student.utwente.nl/sagi/artikel/steelbow/steelbow.html
Now, to topic, you might try ancient Egyptian bows, there's some info in a thread from here;
http://www.primitivearcher.com/smf/index.php/topic,16720.0.htmland probably more over on atarn.net, but I'd need to dig further.
If I come up with anything, I'll post, probably tomorrow.
Frode