I've had cats in my yard, living in a residential area, can't shoot them so ...
I get fly tape that I unwind, carefully so as not to get stuck to it. At the loose end I secure a chicken bone (drumettes work perfect). The end with the loop, I secure to a wire tie (off the bread) then make a hook with the tie to suspend from the rafter in my patio. Cat takes it & the fly tape secures to the cat's fur

Never see the cat again

May not work for your situation, you could also use those live traps & if you catch the cat, take it to the Humane Society ... Just tell them you have no clue whose cat but it's been going after your birds.
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