The extra weight doesn't have to be enough to make it hard to lift their leg to cause them problems.
We tend to think of their legs as scaly, but as scales go, they are rather soft. The flesh underneath can be bruised, and the scales can part to allow tissue damage, pretty easily.
When you take a hard object, put it next to a soft object, and bounce it up and down, wiggle it back and forth, that can damage the softer object. The heavier the hard object, the more likely damage is going to occur.
As a human, you can take the copper bracelet off. You can move it up and down your arm. And you are more likely to adjust the way you move to accommodate the bracelet than an animal is.
As well, your body mass compared to the size of the bracelet is much better than a bird with a leg band (unless you made it a size to fit their toe rather than leg).
An actual comparative size could be calculated by weighing yourself vs your bracelet, then a chicken vs the proposed leg band. I think the size you propose would be analogous to you wearing a metal bracer from a set of armor.
But aside from that, I agree that natural materials are a lot nicer.
I've always wanted to create something with leather, that has a small metal tag on top of part of it with the identifier. Copper would be lovely for that.
