Heavy-gauge bare aluminum wire is also sold in home improvement stores, in the area for chain-link fence materials. It won't rust, and is less expensive than copper.
For fences, the aluminum wire is used for securing the fence to uprights and cross bars. It usually heavier gauge (often 9 gauge, or 0.1144 in. dia.) than the short hook ties also sold for fencing. The 9-gauge stuff should be stiff enough to work for rhizome clips.
I looked into wire and ultimately just got the BetterGro s-hooks for $2-3 for a pack of 3. If you have a lot of orchids, the other options are more cost-effective, but if you only have a few, I found these to be just fine.