If I’ve understood correctly:
Your son went back in on Monday after half term with a cut on his eye caused by his sister, but told school it was done by your partner.
Then on Monday evening he tripped over a stairgate, went in on Tuesday and told school your partner pushed him over?
The school are not “out to get your partner”. Your son has made two allegations in two days; they have responded appropriately by making the referral. They had no choice.
What can you do now? You can engage fully with social services, and show them that you understand that no-one is targeting your partner, that you understand your duty as a parent to keep your child safe.
You say that multiple professionals have said it is due to being in a flat. Why do you already have multiple professionals involved, is there something else you’re not telling us?
If you ARE unsuitably housed with three children in a small flat, you could ask SS for support with a housing referral. And please, please, if you haven’t already done so, get some reliable contraception in place.
You really need to get to the bottom of why your son is saying these things though. He doesn’t sound like a happy little boy. Perhaps school and/or SS could help you put some strategies in place, or support him directly.