I hunt predators (coyotes,bobcats) with my Olympic arms M4 carbine topped with a red dot scope. I have always used the good old 55 gr. FMJ, I find that it does plenty of damage, because its a light, fast moving, tumbling round. When it hits the target it doesn't fly straight through, it starts to tumble nose to tail as it goes through the body cavity. They also dont damage the pelts too badly, unless you hit something in the shoulder at a closer range. The one Bobcat I have shot with mine got destroyed after I unloaded a 10 round clip on it (got a little trigger happy) after it came sneaking in to the call and just about jumped right in to my lap.
With a 1 in 9" twist you're going to want a little longer bullet. I'd go with the 55 or 60 grain hornady v-max. I really like the 40 grain but that really won't work well with that fast of a twist. You'll get an exit with the bigger bullets, but they wont get away. The knock down power in a v max is something to behold. You could try smaller bullets if they're not loaded too hot. A bobcat hide is too precious to waste so if you could go without an exit that would be ideal.
Bring them in on a string. I love this term, if you want to be successful on calling in Elk, try not to just sound like the other caller. We all laugh at each other when we are practicing Elk calling at home or while camping. When I first started calling in Elk, I had no idea what i was doing and it's funny how many I could call in. The biggest Bull that I have called in to date was actually my first. I was making cow calls and screwing it up big time. But it's obvious I didn't screw up...