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.
Here is an inexpenive and effective deer feeder I made recently.
No batteries to go dead, no hoisting it up in the tree, plenty of capacity, very easy to fill - so far, I love it.
I have made two of these and they are easy to build.
An empty 55 gallon drum, some 2x4's, some PVC tubing and fittings, a little bit of hardware and some paint - and you are ready to go.
I designed it so you can stand on the lower rung and fill it - the top is about 6.5 feet above the ground.
It holds about 375...