First, the legal answer. I don't think she could "break the lease" based on what you have written here. Now the practical answer. The more important question is why you would want to stay there under these circumstances. There are lots of problems that can innocently occur in your situation. If something goes wrong, there are lots of more problems that can occur not-so-innocently. It's your life and your situation, but you may want to think real hard about whether or not staying where you are is a good idea for you, for your new wife, for your children, or for your X's mother. I think there is nothing to gain by waving a red flag in the face of an angry bull. To be a little blunt, I don't think you are far enough away from that bull. But like I said, it's your life and your decision to make.
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Your lease is a legal agreement between yourself and your landlord, and each of your rights and obligations are spelled out in the lease - so you need to look at the language of the lease first to determine whether she is barred from entering. Also, many state laws give a landlord the abiity to enter leased property with reasonable notice or if there is an emergency. As for your custody agreement, that is a legal contract between you and your ex-wife. Your remedy, if your ex-mother-in-law sees your kids, could be only against your ex-wife - not your ex-mother-in-law, who isn't a party to the custody agreement. In other words, it's not enforceable if your ex-wife has nothing to do with the circumstances under which her mother saw the kids. The legal means by which to prevent your ex-mother-in-law from seeing your kids under any circumstances is to get a protection from harassment or abuse order from the court, which requires a showing of harassment or abuse. I agree with the first lawyer's answer too - if you really don't want her to see your kids, don't rent from her.
Legal advice depends upon the particular facts of a given situation, please use my answers as general information but not legal counsel.