Would there be a Halachic problem of "T'zar B'alei Chaim", when using a glue trap to catch a mouse?

a frum rabbi who runs a pest control company in another state once told me he finds glue traps with a cookie in the middle to be effective at catching and killing mice.

This is expanding the topic a bit, building on thr matter of responsibility for the suffering of animals. Maybe it will lead to some clarifying insight. Anyway, I have a very strange cat. I got him for $10 for my daughter from a cheapo dive of a pet shop, for my youngest daughter, so I don't know his origins. But he looks "foreign and is both incredibly affectionate and, to the point here, a dedicated killer. He can even be very aggressive to us. We have to read him carefully. I don't know where he could find all the mice and rats he brings home to eat. He doesn't like rat tails though. I have wondered if I am responsible for his killing, and if I am wrong to allow him to do this.

I don't think you need to worry about it at all. It's all natural that animals would hunt other animals.

Thank you Rav. That's a relief. I do try to rescue creatures from our amazing hunter, Pepi. But he is so driven to hunt and so good at it that my success is limited.

I've had some thoughts, following your reply, of our own animal soul and the need we have to deal with it.