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.
Everyone has different ideas about trophy pictures. Looks around and you will see "long armers," "set behinders," "big cheesers," and "blood covered brutes." I have a few tips to keep your trophy pictures from being laughers or worse yet just plain gross.
#1 Clean it up. Try to avoid bullet holes, gut pile, mass amounts of blood, and the tongue sticking out. You don't want your big buck to look like a angry 1st grader with his tongue out or a MMA fighter during the 3 round.
#2 Long arming...