What should I do?

Okay, this is a long story. I have two neighbors, which are also my so-called friends. Well, when I moved here in 5th grade, I met my neighbor, becky. I always thought she was really mean. She just had that atmosphere where she made fun of everyone and complained about everything. But my mom wanted me to hang out with her so I did. I only played with her when she called me, and usually whenever she did call and we hung out she was so totally boring. But two years later, a new neighbor moved in and her name was kate. Becky and I were hanging out the day she moved in, and we both met kate together. We thought kate was cool, and she definitely didn't have that weird atmosphere that becky had. Well, we were all hanging out until two months later when kate got a fundraiser in school. Kate asked me if I could go fundraising together and becky's parents didn't allow her to go. So me and kate did it together, which I guess brought us closer. After a few months, becky started getting jealous that me and kate had such a strong bond, even though we really didn't and she hung out with us just as much.
During the summer, becky started talking bad stuff about katie when just me and becky were hanging out. One day, when katie and I were hanging out, katie told me a bunch of this mean stuff becky was saying about me. And I knew then that this whole friendship thing was going to be a problem. I knew what becky was like, because she usually complains about her "bff" all the time. But then whenever all three of us would hang out, becky would get really quiet and ignore us. One time, we were all hanging out in my room and me and kate went to get some chips. We asked becky if she wanted any but she said no. So while we were out of my room, she locked us out and threw our shoes out my window. Then we tried to get her out for fifteen minutes, but she wouldn't come out. So finally we left to get my mom but when we all came back becky was gone and went home, I guess.
But after that nothing changed. The next day she was totally normal but she did have that weird mad tone in her voice and she started complaining about us right in front of our faces. Finally, both me and kate tried to tell her that she was being mean but she just went home crying and told her mom. Her mom told my mom we were saying mean things to her (even though we didn't) and I got in trouble. So both kate and I finally decided that our friendship with becky was over.
A year later, kate and her started being friends again. And it was hard for me and kate to hang out. So I started hanging out with both of them at the same time because kate said she had changed. But just two days later, we were all hanging out, and that weird feeling was around becky again. Becky was being totally mean to us just like before, and then she tried to make us sound jealous of each other. Both kate and I knew what she was doing at once, but we didn't say anything mean to her. Once she left, I told kate that I can't do it. I can't be friends with kate and becky at the same time.
So I just want to know, should I just ignore becky and hang out with kate (which will be hard because kate is still going to hang out with her) or should I talk to becky and tell her that I can't be friends with her? What sucks is, since we are neighbors, her mom would tell my mom and I'd get in trouble. Grr.

If you actually read all this, thank you so much!!!

I KNOW HOW IT IS 2 WRITE SOMETHING LONG AND NOT GET ANY ANSWERS!!! I think you should tell Becky your feelings and hang out with Kate.. when I first read this I thought my neighbor wroth this and used code names I have such a similar situation.

IM KATE IN THIS STORY..

. you shud realize that if becky liked you it cud of hurt her a lot to c you wit someone else so that is why she was acting bad.