I believe an investment in a screen door for your back door, and
keeping the indoor cat an indoor cat will be the least grief and
heartache for all involved.

Besides the problem of loss, there's the question of fleas and
the diseases (like heartworm) they bring. Then there is the
problem of parasites and other diseases acquired from eating
wild-caught birds and rodents. And injuries inflicted by potential
prey as well as rival cats.

Failing that, the people who make the 'invisible dog fencing'
now offer a similar system for cats.

It can be used in the back yard--- or just across the back
door.