Here is my main strategy. I don't think it works on hard, but it does work on the other difficulties.

There are six cats. 123 on the left, 456 on the right.

Wait until Fooruman unleashes some tube turtles. Let's say he does it on the left. Grab cat 1, go up to the top, shoot. Fall down. Cat 3. When you are just below Fooruman's blue platform, jump off and grab cat 4. Shoot Fooruman. Cat 6. Shoot Fooruman. Cat 5. Cat 4 as soon as it respawns. Up to the blue platform, cat 3. Shoot Fooruman. Cat 1. Shoot Fooruman. Cat 2. Cat 3 when it respawns. Up to the blue platform, cat 4. Repeat.
This doesn't always work, and can be very easy to mess up, but it's what I use. Plenty of room for discovering other patterns, of course.