The neighbor doesn't sound like someone I would want to continue being on friendly terms with. Sounds like a classic case of
JEALOSY to me. If that is the case, as long as this neighbor knows that it bothers you she will continue calling your little one a rat. I would stop the drinks, cookouts, and tool sharing and simply ignore her. Having a realtionship with a less than kind neighbor is worse than not having a relationship with neighbor at all.
I guess it's easy for me to talk, my nearest neighbor is 1/2 mile away - as the crow flies
!!!!!