Status of child born in UK to refugee father and EU mother

I currently have limited leave to remain as a refugee and my girlfriend (EU national) and I just had a baby who was born in the UK. My girlfriend has been in the UK for a year.
Is my child entitled to a convention travel document on the basis that I am entitled to one and if yes which forms and documents will I need to get this?
Do I need to apply for leave for him first or can I just apply for the travel document.

The child will qualify for mothers nationality, or would he not?

I think the child will qualify for his mother's nationality but it will take over 6 months to get a passport for him and we were looking to travel before then which is why we were hoping to get a convention travel document for him.