When is the right time to let my kids meet my ex husbands girlfriend?

My ex and I have only been divorced since December 2010 but he started dating this girl while we were in process of a divorce. All the while telling me he still wanted to be married to me. I am almost sure he is a sociopath as he has no guilt or remorse about the 4 affairs he had while we were married, lies about all kinds of things, important or not, and has major infidelity issues. He seems like such a great guy and you would never guess upon meeting him that he had any issues as he is very charismatic. This girl is 33 and has never been married nor does she have any kids. I have been able to show her proof that he has lied to her countless times and that he has cheated on her already. Proof as in text messages and the like. Despite these lies and his infidelity to her they are now engaged. My best guess is that at 33 she is so desperate and loves the idea of an instant family??? My ex never wanted my 3 kids, 4 and under, to know I had kicked him out after second to last affair (went on while I was pregnant with my 3rd who is only 9 months now) so he always acted like he still lived here and acted towards me like we were still married so the kids would think we were. He was very angry at me when I told the kids we were not together anymore. Until a few weeks ago this is how he acted in front of the kids around me, using pet names, hugs, and telling the kids he loved me. (keep in mind I knew full well this was a show for the kids!) But now he all the sudden wants the kids around his girlfriend and for them to be an instant family. Being 4 and under they are very confused as to what is going on with us since he was so reluctant to be honest from the beginning, we are going to have to move out of our house since I can not afford to live here on child support alone and can't get a job that pays enough to put three little ones in daycare. So lots of changes already for these children and he wants to add yet another house for them to go to (he currently takes them to his parents house where they are comfortable), and a new woman in their lives. I am stressed about all that is taking place for my kids and never in a million years thought this would be my situation. I just don't know about letting this woman who has decided to stay with a cheater and liar and get engaged to someone who is obviously troubled 5 shorts months after his divorce is final and she knows all about all his affairs while we were married as well. One last huge piece of information is that my ex insisted when we talked about divorce that no man be around my kids if I dated so in the divorce decree it states that no person of the opposite sex involved in an intimate relationship may be around the kids. He is wanting to violate this. Any answers to my questions or insight into what this girl might be thinking would be awesome. Thank you!