I'm so very glad you've decided on the course you have. It would be a tragic and unnecessary waste to put him to sleep for following his instincts. I realize it's painful for your neighbors and they lost a beloved pet, but I think it's over the top to be pressuring you to have your pup put down. Honestly I also feel they shouldn't have been letting their cat run around unsupervised outside either, there are a lot of dangers of letting a cat wander the same as letting a dog wander.

It has to be a truly difficult situation for you, I wish you and your dog the best with all my heart