Unfriendliness to Less Observant Jews

Dear Rabbi,
Is there any Halacha that forbids strictly religious Jews from socializing or even returning greetings (such as “hello” or “good Shabbos”) to less observant Jews? Well, many people who live in our community apparently believe there is.
My wife and I are Orthodox Jews, just not in the same way as most of the Orthodox Jews of our community. We don’t appear outwardly religious. I am not a full-time kippah wearer and I never wear black hats. My wife wears short sleeves and pants and shorts a lot of the time and doesn’t cover her hair. Both of us wear T-shirts and jeans most of the time.
We are members of a shul that has a low mechitza and where most people drive there on Shabbos and are fully accepted, although we walk there. We keep kosher, but we don’t follow strictness like Cholov Yisroel, for example. We enjoy a lot of secular entertainment. Most of our close friends are non-religious Jews.
The street we live on has mostly those who are supposedly more religious than we are. We try to be friendly with them. We consider them our fellow Jews, even though they are different. And they ostracize us over these differences. They return greetings like “hello” and “good Shabbos” with silence and sometimes dirty looks as if we invaded their personal space. The only time any of them make conversation with us is when they are angry about something, such as when our 2-year-old wanders into one of their yards. One time when that happened, I told the guy “you are my fellow Jew. You should be nice to me.” He just walked away with a mean look.
I once took a shiur when I was growing up in which the rabbi said it is a mitzvah to say hello to another. And if someone says hello to you and you do not return the greeting, it is as if you have robbed that person. The rabbi said this as if it applies to everyone you greet or who greets you, no strings attached. I was highly influenced by what I learned.
I am wondering what you think as a rabbi. I know you might tell me that I should just become like all my neighbors and the problem will be solved. But that’s not who we are, and if we ever became that way, it would all be for show and would not be in our hearts.
My question is, are there Jews who are so religious that they consider it permitted and maybe even required to play deaf to a fellow Jew’s greetings or any attempts to socialize? And is there any basis for such behavior in Torah law?