I really don’t think the kind of metal or wire you describe is suitable for a bow.
I’ve looked a several sets of instructions for homemade bows and they all suggest using cord or string. One of the reasons, I suppose, as to why it is generally referred to as a “bow string.” :-)
For example”
This guy used kite string
And this guy used reverse-wrap cordage
From his instructions:
Bowstrings
Bowstrings are made using the reverse-wrap method of making cordage that I described in Mother Earth News Volume 79, 1982, only you will be using sinew in most cases. For bow string, what I do is a double-reverse wrap. By reverse wrapping more than ice the length of the bow, then loading the cord in half again and reverse wrapping it, you will have a very strong and durable bow string. . Plant fibers can also be made into bow strings the same way, but they are pot as good. Plant fiber bow strings, however, will not stretch like sinew bow strings in foul weather. Velvet leaf, dogbane, and nettles make good bow strings for wet weather purposes.
I have no idea what double-reverse wrapping is, but you want to give it a try, I’m sure it’s very easily googled. :-)