On the other hand, regarding what I wrote, I was rethinking this over, and realized that you’d be sending an indirect message to them that you want to get closer and welcome a part of them into a part of your lives”
I have given gifts to MANY people to whom I have no reason or wish to be closer in other than a professional way(my kids’ teachers or rebbeim, my mailman, the wait staff whom I tip, etc.) Gift does not = welcome into my personal life. It is ALWAYS a good policy to be on cordial terms with one’s neighbors, and that does not mean you must have them over for dinner.
Remember the story from the Holocaust of the frum yid whose life was saved because he always said good morning to his German (Nazi) neighbor. When that neighbor was sending people to the right or to the left, the yid recognized him and say good morning to him as he had always done. The nazi sent him to life and not to the gas chamber.