I think you need to speak to a solicitor who knows the system, really. A specialist in children's law. This is such a complicated situation.
My thoughts, on what you've said: one problem is that as a homeless person with a child, she's priority in housing terms, so GS is an asset to her, bluntly, if she's kicked out - she has no such priority otherwise. And SS have - necessarily, given the huge intrusion on family rights of separating parent and child - very low standards for minimum levels of care. In terms of the OP offering better care within the family, well, sure, but parents have a presumption in their favour. So if DD is angry enough at the SC being involved behind her back, she could use the best weapon at her disposal, too, which is taking DGS and making a show of working with SC to placate them long enough to get them off her back, and then OP would have a battle on her hands just to stay in his life at all. I do understand why you're so worried. (I say that as someone who has requested Social Care involvement for my kids due to disability before and had Safeguarding sent around instead, as we were in an EHCP battle - they were bloody wonderful. Wrote us a report that lauded us to the skies, and were scathing as could be about the SEN provision from the LA - nobody could have more reason to know how brilliant social workers can be! It's not the social workers I see as a problem here - it's the DD's reaction to their being brought in over her head, and the law being heavily weighted towards parental rights. If she could put her son first, and react like an adult, the problem wouldn't arise).
There is another alternative - to offer to put a deposit on a 1 bed or studio flat for her, and she can leave DGS with you, but you need PR, in case of accident, for schools and for stability, so she needs to consent to a Child Arrangements Order in your favour as part of that plan. You never know - she could get it together in future, and he needs contact with his mum, so you could maybe agree to Sunday lunches, days out set in the CAO. Even if she doesn't bother to do it, at least you've tried to keep that connection for him, and if you keep a record of her contact efforts, you can evidence it if she just drops him, too, and at the least insist on a slow and consistent return to her if and when she does get her act together later - if she's not got that act together, she won't be consistent with a staged return, either. You get all of this written up formally, and keep a paper trail, and then she would need to launch her own legal action to get him back - which would mean a Judge oversaw it all. From what you say, how likely would she be to do that? And if its' a private arrangement, and not a state one, she'd have no right to legal aid to get it changed - she'd agreed it was in his best interests. And to get her to agree to it I think focusing on her self interest would be best: a house of her own, contact with her son (also in his interests) so she feels she's not losing him, yet none of the hard graft and unselfishness that caring for small kids necessarily requires.
I'm afraid I don't know if you can still get what used to be called a Residence Order in your favour which automatically confers PR and states that your home is where he lives - I know it was, but the law changes so fast. You need to ask a family solicitor for advice really.
Your GS needs stability, and you need to ensure he has it. In your position, that's what I would be looking to do. I'm sad for her but she's not a child, he is, and you have to focus on him.
I realise you won't get the foster care payments that might be available otherwise, but it does reduce any chance that she will hit back at you from spite and take him just to make a point. And if you have PR and a Child Arrangements Order, then you'd get child benefit and top-ups to Universal Credit, I would imagine.
Social Care would be the best option only if you are quite sure your DD would rather leave him with you, even if it meant she had to sort her own housing, and even if she saw that as you having 'won'. Only you can know her temperament - perhaps I'm wrong, and she's not prone to immaturity in that specific way along with the rest. If you're certain she won't do that, and can sort her own housing out (or if you think if you offer the 1st month plus deposit she will go even with Social Care involved, and him staying at yours) then Social Care would be best because, bluntly, your grandson is going to need one hell of a lot of support and accessing that support is easier when on Social Care's radar.
I'm so sorry. It sounds heartbreaking. But he's been exposed to God knows what in his short life and he needs to feel safe and loved and wanted - and with a routine that is so key to that, for small children.
Again, I'd talk to a specialist solicitor about next steps. They will have seen a variation on this theme more often than they would ever want to, I'm sure. If they tell you to get Social Care involved, having considered all the angles, then you can feel safe about doing so.