The Compiz Fusion worked just fine before I installed the updated ATI drivers. >.< Same results as the Radeon 2400 Pro.
Since it worked before the ATI drivers, it is possible that ATI's driver is blacklisted by Compiz. It could be any number of things, but try this:
grep "WHITELIST" /usr/bin/compiz
You should see a line like WHITELIST="nvidia intel ati radeon i810". If so, that means that it may refuse to work with the ATI provided driver. To fix this, edit the file /usr/bin/compiz with an editor (e.g. nano). You'll need to be root obviously. Then add fglrx to that list so it'll look like:
WHITELIST="nvidia intel ati radeon i810 fglrx"
Then try it again. This was from the Ubuntu Guide
section on "ATI users and Compiz
There are other potential problems, too. You said you installed both the manual install ATI provided package AND the Ubuntu repository package from restricted modules. In all likelihood, they will use different kernel modules and the restricted module package version will conflict with the other one. The kernel driver may not load properly then. Also, there's a variety of Xorg settings that could be changed to the incorrect settings by combining the different install methods.
BTW, also post the output of the following two commands (just to sanity check):
grep -i "driver" /etc/X11/xorg.conf
lsmod | grep fglrx
glxinfo | grep rendering