As he keeps contacting SS himself with ridiculous claims, they are aware of what he is like and by now they'll know that he is just talking bollocks. This will go a very long way in your favour.
See your solicitor, and find out for sure whose testimony the Court will take; will they take info from SS, your SW, CAFCASS, police, WA, GP, HV, nursery, who?
When you know whose reports will be standard in the Court, then collect letters from as many agencies/organisations as you can - letters or reports, saying what they think. Find out what you can submit for your evidence, and then get as much as you can, from your mum saying she's prepared to help (if she is and is able to and you want her to) to an official reportfrom your gp as to your state of health, WA outlining your case and reiterating their willingness to support you, etc etc. Everyone who has a good word to say about you and your care of dd!
This will stop, really it will. And you and dd will be happy.