I've never tried installing the ATi drivers on a live system, but that would no doubt be pretty difficult. You can try a full installation on a USB stick, but that will be very slow (and also kind of hard on the USB stick, due to the high number of rewrites incurred on a full installation). If you have a USB hard drive, installing AVLinux to that will work very well.
Here are a couple of options:
1. Install the ATi drivers on a live booted system:
Download the ATi driver from here:http://support.amd.com/us/gpudownload/linux/Pages/radeon_linux.aspx
If you save that to the hard drive of your netbook, you should be able to run the ATi installer (not sgfxi) from a live boot(it can run in X). After it completes, it will want to reboot; I think you can switch to a tty (Ctrl+Alt+F1), log in as root, and run 'service slim restart' to restart X with (hopefully) working ATi drivers.
2. Create a USB stick with persistence. This would let the 'live boot' system keep your changes through reboots. There is a guide here on the forum for doing that, though it didn't work for me (only tried it once, briefly, a while ago).
3. Install AVLinux somewhere, install the ATi drivers, and use remastersys to create your own custom bootable ISO.
If you use option 2 or 3, you could also install the 3.4-rt AVLinux kernel that is in the 'testing' repos (see the AVLinux kernel repo thread). I would be interested to know if that kernel has better support for your video card.
In any case, it's highly unusual for sgfxi to fail to download the driver packages. That link appears to be dead, and sgfxi is updated quite frequently, so I'd expect it to be fixed very soon. But having to download the drivers every time you reboot sounds like quite a pain...