I'd avoid using wire. I used wire to connect everything to the screen and within 2 months it all rusted and could basically pulled apart. I did a deep clean yesterday and removed all of the old wire and instead used fishing line which is perfect. Fishing line is waterproof, can hold a ton of weight, and you can barely see it. Because I have branches of wood in my cage I just draped the vines around the wood and made a highway system. But there were times where I wanted the vine in a specific spot so I would place the vine where I wanted it and string the fishing line around it and then have someone else knot it from the outside of the cage.
But if you're not going to use wooden branches I would use the dowel rod method.