Whoops, I missed the part about using netting for the top. Yes, then ramblers would be out of the question. You could stick to shorter-growing repeat climbers, which usually max out at about the height of the walls, or could easily be trained horizontally across the walls (and give you more flowers) rather than up and over the top. Aviary netting isn't really an option for some of the species I was interested in for myself, so I just took that out of the equation. And my whole reason for going south is to finally escape frost -- I'm so over winter.
OK, so less exuberant climbers would be found among the classes below:
Large-Flowered Climbers
Hybrid Musks (some)
Bourbons (some, see growth patterns)
English (some, see growth patterns)
Climbing Polyanthas (the less-rampant repeat-flowering ones)
Or, you could stick with ramblers, but rather than training them up, train them horizontally, wrapping around the outer aviary wire. You'll find applicable references by searching for "training roses along a fence" or some similar phrase. You take the long canes and tie them horizontally along the wire. They'll send up lateral shoots along their length, which is where the flowers come from. You just whack back the laterals whenever they get too crazy.
THIS pic of a rose trained horizontally along a wire fence shows you what I mean. The pic was taken during winter, so while not very pretty, you can more easily see the "bones" of the plant.
HERE is another, showing leafless canes -- laterals will sprout along the horizontal length. If you want a rose to grow this way, you can't just get any hybrid tea from your local nursery and expect it to work -- you have to find types that grow long, somewhat flexible canes that lend themselves to training.
HERE is a blog post with instructions and photos for training ramblers horizontally to a fence. If you want something less rambunctious, choose a repeat-flowering climber with flexible canes -- from the classes I mentioned above.
Or just ignore my crazy rose fetish. To say goodbye to some special people I'll be leaving behind when I move, I decided to pick out some of my favorites and leave a gift-rose for each person. This turned into an excuse for plant-shopping, and now I have 41 coming in a month and a half. Probably 15-20 will be moving with me and finding their way to friends' yards back home. Almost all the varieties were bred 100 or more years ago. I'll share pics after they're planted.