They say during the search the AOS dog was aggressively attacked by the family's dog, Sully, on the verge of the road.
Sully, a 6-year-old staffordshire-huntaway cross, was one of the family's four dogs. He was pepper sprayed and shot dead.
The AOS dog is recovering with facial puncture wounds.
Police arrested the 31-year-old woman for breach of bail.
Sully's owner Debra Vanin says the event was traumatic and had taken an emotional toll on the family.
She disagrees with police claims that her pet was aggressive.
She says Sully was shot on the family's back doorstep of their house — not the verge of the road, as police claim.
Waikato west area commander inspector Andrew Mortimore said in a statement the shooting was a "very rare occurrence".
"On this occasion staff determined the use of a firearm was required, given that spray had not worked and the police dog continued to be under attack from the other dog.
He said the dog was shot on the grass verge at the front of the property and then moved by police to the back door.
"Police do not want to be shooting or injuring dogs or anyone, however the actions of our staff are driven by the need to protect their own safety, that of our police dogs and the safety of the public.
"The decision to use a firearm is never taken lightly by frontline staff, who must make quick decisions in dynamic and ever-changing situations.
"This is always a last resort and victim support has been offered to the dog's owner as we recognise this is a distressing event for them."
However, the family says it has not received victim support.
Another of Debra's daughters says Sully was acting on instinct.
"Our boy Sully did what any good guard dog would do in this situation — he protected his family.
"He fought with the police dog — an instinct that any dog on this planet would have."
She believes the death could have been prevented by police breaking up the dog fight or using a Taser.
"Is getting the job done faster more important than taking a minute to break up a dog fight caused by the police intrusion?"
"You may say that the police were just doing their job.
"But so was our boy, Sully."
Debra says the death of her dog was an unfortunate consequence of getting their daughter bailed to their house.
She says the family understands police were doing their job in searching for a wanted man.
