Yea I can't really do the poison thing because of the cats....Also I tried the live traps to be humane and all yet the mice won't ever go in those but you put out a kill trap and it gets one in hours.
I have had a mouse issue twice in the last 3 years and also have a cat. Fortunately for me it was confined to my pantry which is always closed. They found their way into my cupboards but had no way to get out of the cupboards. I put a poison bait trap in there and it got the mice both times.
Now for your situation, they do sell some poison bait traps that are covered so the cat can't get at the poison (the kind I used was covered). But you would have to be very vigilant to watch that the cat didn't find the mouse after it ate the poison and then eat the poisoned mouse.
They also sell some traps that are like a poison bait trap where the mouse enters, eats the poison and can't leave the trap. Well, theoretically it can't leave after entering, I'm not sure how well that works in practice. I would still be very careful that the mouse didn't find a way out and get eaten by the cat.
Finally, if you google "Bucket mouse trap", you can find instructions on how to make a live mouse catcher (add a half foot or more of water to the bottom of the bucket to make it a kill trap). Which should be fine with your cat but requires just a little work on your end. Good luck with your mouse hunting!