Sorry, don't know how the CSA works, as I don't live in Britain (I never went for any maintenance, long story, from a "3rd world" country, basically and gone back)
But I don't think not putting your child's father on the birth certificate equals him not being eligible for maintenance
(correct me if I'm wrong), but legally and DNA are two different things, no?
I think women now can register a birth without mentioning the father? (always could, but not the stigma any more, thank God!)!
I would go for the first, in legal terms, money you can always get, somehow, and you'll be amazed how creative you get as a single mother! 
That child of yours is going to change your life!
I always say "He's the light of my life and the centre of my universe"!!
Actually, a good idea to go to CAB and get all your legal rights sorted! They are the ones who know!