Tiger, you seem very worried about SS not agreeing with your DH's decision to bring his daughter home. You do know don't you that they are not the people to make this decision.
SS do not make the choice about where children should live. They make recommendations when asked to by a court. Right now there is no court to involve them. SS would only be able to remove DSD from your DH's care if they can prove she is at risk of 'immediate significant harm' and they can prove it to a very high threshold.
SS may have sided with the mum and SF in the past when there were allegations from your DSD that she was assaulted by her mum and emotional bullying by the SF, but that should also show you the high level of proof needed to remove a child from a parent. They didn't have it then so they couldn't remove DSD, and they won't have it now to remove her from your DH's care.
SS won't try to approve a move for DSD from the SF because they have no need to. She isn't in danger with SF to their eyes and so they have no need to bother themselves in this case. They have a high workload and wouldn't look to create more unnecessary cases. However this is all irrelevant their approval or help isn't required here your DH can just remove her to his home perfectly legally.
Please, please look into alternative legal guidance today. You and DH need to be very clear with your requirements too, " I am the only living parent and I have PR. The SF and the school are being uncooperative. I want my DD to reside with me. What steps do I need to take to achieve this today".
If you are giving your current solicitor the benefit of the doubt he/she may have believed you wanted to raise the issue of ongoing contact rather than where DSD resides permanently and has rightly pointed out that SS would be unlikely to 'authorise' a change of residence. You need to make it clear that you and DH do not want their authorisation as you do not require it. You want your DSD with you. Now.
You may find you get an alternative response from the legal bods then.