Ah! A cat bow! Now that is a hearse of a darker color!
Here's the scenario. Dogs strike hot scent, cat lights out for the high country. You are packing as little weight as possible to keep up with the hounds. You scrabble up and down rough country trying to keep as close as you can when you hear that one hound with the bell mouth light up. She's usually a gentle bitch with you and kids, but that all goes out the window when they get something treed. Your heart is already pounding from the excercise, but it grabs another gear you never knew you had.
I'll tell you this, English Longbow styles are the most accurate and forgiving of all bows to carry and shoot. And also the last choice for running thru cover, bailing off ridgelines with rocks and scree racing you to the bottom. You might consider a recurved, bend in the handle, sinew backed, almost-horsebow. Your shot will be close range, but you want some power behind it. Osage, or hickory if ou live in a dry climate, inch and a quarter wide and two inches shorter than your draw length (providing you flip or recurve the tips). This could be as short as 50" depending on how long your draw is. Cover the sinew backing with rawhide and a Massey finish to seal 'er tight would be my choice.
I hunt cougar in the Black Hills of South Dakota much the same way, without the dogs. Dogs are illegal here at this time. I strike track on mornings after a snowfall and push like a fiend to try to bay or tree the cats. I carry the bow and three arrows in my left, using my right hand to push treebranches and brush outa the way. The bow is unstrung and I use a wrap of masking tape as a string keeper on either end. That way I avoid having to worry about getting the bow hung up on an obstacle.
I've never treed a cat yet, but there was one I was pushing pretty hard, he was doing 30 ft a jump at one point. But the sun was going down and I had covered at least 8 miles after leaving my vehicle. Lemme tell you, that was one cold and scarey slog back to the old Isuzu Trooper. I ached for 2 weeks.