I think you & your OH have been incredibily brave about the whole situation & by the sounds of it shes lucky to have you both in her life.
DD obviously feels like this for a reason at home with Mum first I would ask exactly why - find those reasons and always make sure she never faces that at home with you guys. One possible way is for time to be spent with child alone shopping park etc another is to have her take part in babies life too obviously not put on to look after but 'help' & play - not left out .... like in a situation baby is getting attention with a play on the mat ask DD to come sit with you - sounds like its possible this happens at home & child is ignored and assumed shes ok shes older!
Going back a bit I would find the reasons of her feeling like this about her sister then maybe talk to her about future family & mildly talk & comfort her insecurities too - lightly suggesting how it
won't be which will be what makes her feel like this now.
Also one thing to consider - is DDs Mums partner babies Dad?
Sorry if Ive misread any of your post hun
