Importing non-indigenous species for pest control has always been a horrible idea in the past. One good example of this is the cane toad in Australia. It ate or drove away all the native species and started to destroy the crops. It became so bad that they had to start culling the toads that they released.
The import of non-indigenous species will always have unintended consequences. Specifically, ask people who favor such importation what they plan to do when that pest is successfully eliminated. The things you imported to get rid of them aren't going to say "Oh, job well done, we'll just all die now," they're going to turn to other foods to survive. Good luck with that.
Introducing foreign species has been shown to cause severe environmental damage. It is never a good idea for humans to start manipulating he ecosystem because there can be consequences they cannot anticipate. Other steps can be taken to control pests in an area, such as hunting or the use of chemicals.
Non-indigenous species can quickly overpower those already in place in a certain area. A current famous example is the Asian carp that is quickly overtaking northern waters and the fish could make it to the Great Lakes. Another example is how Europeans overpowered Native Americans from Mexico to Canada and took their land as part of so-called "manifest destiny." Non-indigenous species for use in pest control rarely work. Killer bees in South America are another example, although those insects weren't used for pest control.
It is not a viable expansive approach to import non-indigenous species for animal pest control. Any species introduced into our ecosystem would ultimately destroy that ecosystem. If we did introduce a new species for an already common pest, how long before we have to introduce another to keep that one from superseding the other one. It is a never ending cycle and the ecosystem is very sensitive.