My DH's ex is also high-conflict and chronically circumvents court orders; she used to deny contact and use moving as a tool to keep DSS away from DH. She still tries hard to manipulate DSS into choosing sides. DSS is about to turn 13 and DH has had sole custody since he was 8.
I feel compelled to say a couple things you didn't really ask about:
1- Kudos to you, for having a back-up plan. It sounds like your DP has good reason to ask for custody, but he will be at some disadvantage. If nothing else were an issue, he's still the male parent. All along, he's had the choice to fight against Mom's court-order violations or try to keep peace for the sake of his son. Both choices are defensible and respectable. But the side-effect of the path he chose is an absence of contempt findings against Mom. Also, your DSS is old enough that many courts would want to know his wishes and factor that into their decision. And it sounds like Mom has already turned him against your DH? There are sound counter-arguments to each disadvantage I've mentioned, so your case is by no means hopeless. But, rather than convincing yourselves winning custody is a "sure thing", it sounds like you've planned to at least define your DP's summer parenting time so it's enforceable, if you don't win custody. That's a good plan.
2- Request a custody evaluation. They're expensive, but in a case like ours (and perhaps yours) invaluable. The judge can't simply listen to your descriptions of everything that's going on and what you think is best for this child. (By "you", of course I mean your DP and you, to whatever extent you are involved in court.) The judge is restricted by the wording of laws, orders and guidelines and somewhat bound by them, even if they prove inadequate for your situation. (High-conflict parents like your DP's ex may find every exploitable loophole in a court order.) The judge must also consider precedent and standards of practice (which also may insufficiently address the slippery things your DP's ex does). Many, many things that seem obviously relevant to a child's best interest won't be admissible in court because of strict rules of evidence. Even your DP and his ex's testimony will be colored by which questions the lawyers think to ask (and fail to ask); how quickly her attorney cuts off your DP while he's answering; his attorney's reluctance to seem like he's bullying Mom on the stand; both attorneys' desire to maintain a positive relationship with the judge, since they'll have to keep appearing in that court long after your case is over, etc., etc. The judge also has to be concerned about getting overturned on appeal. He/she may know, in his/her heart, that your DP should have custody, but he/she needs enough hard facts, precedents and "legalese" to back that up, on paper.
Whereas a custody evaluator will spend individual time with each person involved (including you); let everyone speak freely; observe how everyone interacts with the child; likely do in-depth personality testing that may show, for example, if a sweet, honest-seeming mom has tendencies toward pathological lying... A CE's in a much better position than a judge to tell if your DSS is being manipulated into siding with Mom and can advise the judge that DSS may not have a good sense of what's best for himself. And having a CE testify that it's in DSS's best interest to live with you takes a lot of weight off the judge, in terms of worrying about being overturned on appeal if he/she gives you custody.
3- Don't be too quick to sign away so much of your "down" time with DSS, if you get custody. If, as NCP, your DP gets half of weekends, holidays and summers, why does he plan to offer Mom basically 3/4 of weekends and "most" of summers, if he gets custody? If there's a good reason (like his work schedule), fine. But if he's just doing it to sound generous in court, or in hopes of softening the blow for his ex (and therefore his son), reconsider!
>> Having DSS spend a majority of his total days with you may not help the dynamic where he sides with his Mom and rejects your DP, especially if most of the days he spends in your home are school days (between school, homework, extra-curriculars, chores and/or a part-time job, and having to get to bed at a decent hour because he has school the next day, "family" and "fun" time will be limited) and he'll either be out of school or on an extended vacation, EVERY day he spends at Mom's. Your family will need your share of weekend and summer days with him, if you have 100% of the school-week responsibilities. You might consider offering Mom one weekday per week (if she keeps her residence close enough to exercise it), but only EOW and half the summer.
>> If Mom is flaky and moves a lot - and especially if she's furious with your DP for getting custody - at some point she may move too far away to make it convenient for DSS to hang out with his school friends, during her parenting time. That would really stink, for him - and also ensure that every possible second of weekend and summer time he has with you, he'll want to spend with his friends. Your chances of getting some enjoyment time with him, as he becomes an older teen, are better if you have a reasonable amount of weekend and summer time.
But as to your actual question... He's simply too old for you to shield him from everything. There's a difference between saying snarky, mean things about his mother and telling him the truth (within reason), even though - by Mom's own choice - the truth is not flattering to her. Plus, if you know he's hearing false, negative things about his Dad, it's not healthy for him if your DP lets that stand. Your DP isn't selfishly defending himself. He's defending his son's need to believe the father who contributed to his DNA is a decent person!
I think your DP should wait until he's actually filed something in court. That way, DSS won't feel it's his job to argue his Dad out of doing it. It's already done. Plus, it won't matter if DSS blabs to his Mom. She's going to find out, anyway. Immediately after filing - before DSS hears about it from his Mom, before she's even served - he must sit DSS down and say something along the lines of, "I've decided to ask the court for custody. I know you might be upset about this, or worry how your mom will feel about it. But I'm not trying to keep you away from her. I love you and care very deeply about what's best for you. As the adult, the parent, I have spent a long time considering all the angles and I believe it would be best for you to live with me. I believe it so strongly that I'm willing to pursue it, even if it makes you angry with me. I still love you. Of course your Mom believes it's best for you to live with her, so it will be up to the judge to decide. I didn't discuss this with you first because I don't want you to feel it's your decision. You shouldn't have to choose between your parents. You ought to feel free to love us both. We both love you. You're old enough that the judge might ask what you want, but that won't be the deciding factor. The adults have to consider numerous other things that affect your best interest. You're allowed to say that you don't want to choose, you don't want to declare a preference. But if you say you want to live with one of us and not the other, you don't have to feel guilty. It won't be your fault that you live away from the NCP. It's your Mom's and my fault, that you can only live with one of us at a time. No matter what you say or how you feel or who you get mad at, this cannot be your fault."
He needs to be prepared to give his son some details of what life would look like, if he lived with you. Where would he go to school? Could he continue whatever activities he's currently involved in? Who would be home, when he's home? Would your household rules or his chores change? When would he see his Mom?
He also needs to be prepared to answer, "Why do you think I should live with you, Dad?" It's fair to tell him that, as things stand, your DP has trouble exercising his parenting time; he doesn't always know where DSS lives or how to contact him, etc. You don't necessarily have to go into Mom's presumed motives for creating those difficulties. The fact is, they exist. DSS already knows this. He can make his own assumption about why his Mom is the way she is.
You never know. He might have some quiet resentment over why his Dad has kept quiet this long and not fought to end some of the chaos in his life, with his Mom. He might actually be glad, to see his Dad standing up.