I am having a hard time getting my heart in the right place about a situation that has reared it's head recently.
DH and I have been living together for a year and a half. He has 50% custody of a 16 yr old (boy) and a 12 year old (girl).

We got together after he had separated from BM, she was the one to pull the plug on the marriage because she thought the other guy she had been having an affair with for 5 or so years was ready to dump his wife and make a legitimate go with her.

Though BM was (and still is) a total asshole to DH, just straight up narcissist cruelty, he decided to go through mediation instead of court to settle custody/support/property issues. He pays her alimony and child support faithfully. Their divorce and settlement was final April 2016. Part of the settlement allowed her to live in the house they still both own until October 2017, after which time it was to be sold, debts paid off, and proceeds divided.

She has made no effort to find another place to live up to this point. When an offer was made for the house recently, she agreed to the price, but tried to get the buyer to agree to rent back to her indefinitely at a rate that would not cover the cost of the loan, so she could "find a place for her children and her to live." When the buyer declined, she decided (without consulting DH) that the sale was off until she could figure out housing. Given her total laziness and general attitude that DH should be responsible for her well being because she sacrificed so many years being married to him, I doubt she will get off her ass and find a place unless forced to. He was collecting documentation of such for a return to mediation to either get her to agree to the offer, or take her to Court if that failed.

Then she went nasty. Out of nowhere, the SD tells DH she is angry with him last week. She requested to see her counselor prior to telling him why (not a bad thing). So he takes her to the counselor. She then has the mother of all meltdowns later that night, telling him he is responsible for making BM move, that he does not care what happens to BM. Essentially, her mother told her it was her dad's fault she had to get rid of a family pet, a car, and he didn't care if BM was homeless. SD is refusing to go on a prescheduled camping trip because her older brother can't make it, and suddenly, she "doesn't feel safe". There has never been an incident to make her feel this way, except for DH telling BM it was inappropriate to ask the buyer of the house for contingencies she had not previously discussed with him, and that the kids had a house (ours), so to paint them as being at risk of being on the streets was not honest.

BM isn't as engaged with the almost 17 year old, I believe because he is not as emotionally vulnerable and because he won't generate income for her much longer. I really feel for DH and have told him I think he needs to pursue legal adjudication to get her the hell out of the house, and have it sold no later than October 31. In my opinion, the longer she lingers, the more drama and alienation tactics she can dump on the SD. He is unsure about that approach, fearing it could enflame BM even more, causing harm to SD. Even though she is about to receive the 70K payout from his retirement, he has seriously considered helping her find a place in hope that she will stop harming SD emotionally. I think it will have the opposite effect, empowering her to use that tactic every time she wants or needs something she feels he is responsible to her for.

So essentially, BM is playing him with the old playbook from their relationship, using the SD as a weapon to get what she wants. I am trying to be supportive, cause it is awful to see him go through this, but part of me is losing respect for him, watching him buy right into this lady's game. Meanwhile, both his kids spend most of their time at our place (even on her days) because we have food in the house and it's a no drama zone.

What approach would you take if you were in his shoes?