One of my cats; a semi-feral female kills shrews, she never eats them though. She's made it a point to kill every shrew she encounters, and there have been many. I'd suggest a cat, a female or male kitten, say offspring of a feral female or a barn cat. The cat's mother usually teaches her kittens to hunt at a certain age, if the mother is a hunter its likely that her first-born female will be taught to hunt at a young age. If the kitten is taken away too early, before say three months old, she won't have been taught to hunt, but if she's with her mother for longer then she will kill shrews and rodents easily. The problem is always if the cats have seen their parents hunt, and some even if they have, won't hunt except to eat. Actual true hunting cats are not easy to find, many will have the desire to hunt but not the ability, many will hunt for food only. I'd suggest that you find someone local with a barn cat with older kittens, and actually find out if there is a true hunter among them.
I've lent her out to a couple of friends for a day or two, when they had rodent problems. They said afterwards that it was like a bloody battlefield and they haven't seen any mice since. Unfortunate that you can't have a cat in your present circumstances.
Now understand why my cat won't eat them:
Toxin
The
saliva of the northern short-tailed shrew contains a
kallikrein-like
protease, used to
paralyze and subdue its prey. The toxin is strong enough to kill small animals, up to sizes somewhat larger than the shrew itself, and results in painful bites to humans who attempt to handle the shrew. The venomous saliva is secreted from
submaxillary glands, through a duct which opens at the base of the lower
incisors, where the saliva flows along the groove formed by the two incisors, and into the prey