Will chicken wire make a good trellis for some morning glories and english ivy?
Chicken wire won’t hurt your plants, but it might very well look crappy in a few months, as you say. You could regard it as being temporary. Another thought would be to run the overflow pipe from your air conditioner out to where your dog can access the water, maybe flowing into a bowl. There’s no sense wasting water, and if your dog wants it, it’s fine for her. Then, once she learns that the water is now accessible, you could pretty much do what you want. I’d go for the chicken wire if it’s handy and plant morning glories, because they will quickly cover it. I’d never in a million years plant English ivy anywhere, because it’s a foreign invasive thug that plays no useful part in our ecosystem, but that’s a personal issue.