What about a bike handle bar grip tape? It's spongey tape that you can fix your bicycle handles with. It wouldn't get hot and should stand a bit of the elements.

The best thing for full sun is exterior shading of some sort. It doesn't look ratty, it's just essential. I think I would have a look for your options at bunnings or whatever. Then put it to her in writing what it needs for fixing to the house properly. Then it's up to her, but if you are buying the shade and taking it when you go, then she shouldn't complain. I am guessing there is no property management involved - just a private arrangement?