Downsides of using an existing length of wire as a shunt
There is no downside for your job. You just have to make sure you organise it as a proper 4 wire connection. i.e. the two sense wires have to attach to the wire itself, i.e. be inside any connectors.

You can strip the insulation off the current carrying wire, and wrap the sense wires around it.

You can also get a big chocolate block type screw connector, and pass the bare current carrying wires straight through the whole sleeve. Then slip the sense wires in, and screw it up. 
Only use these chocolate blocks for the sense wires, the current connection must be beyond them.

Stainless wire. or a thin strip cut off some stainless sheet is perfect. Stainless bicycle gear/brake cables are common in my workshop. Actually I have seen these used as shunts in commercial equipment before.

Copper has a poor temperature coefficient, but you aren't trying to make a precision measurement, so it doesn't matter. 

Interestingly, all the pure metals have about the same tempco 0.4%/deg. (so you can use any pure metal wire to make an RTD sensor)

Resistance alloys are 20x better, but ordinary stainless steel is 4x better than copper, has much higher resistance, so can be shorter. 

That said, a long , low resistance, piece of copper wire won't self-heat, so its temperature will be more stable, and the nett change less, than a short piece of stainless that gets hot.

Ordinary mild steel is not very different to copper.