Stay away from the faster shells and use a rifle with a longer barrel. That is about as good as you can get without using a silencer which is an expensive option, or depending on where you live not an option at all.
Unless you use subsonic rounds they will all make crack when they break the sound barrier.
For quiet cat control I like to use a bow with blunt tips

. Best thing I have used on something that size is my .17HMR, it's not very quiet though.
I say a nice full size rifle with 40 grn target loads should work well. Shorts would be better yet if you can get them to shoot well. I haven't had very good luck them myself in the accuracy department.
I could offer you some other advice, but we can't talk about it here for legal reasons.[

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Paul