I am in the exact same position with my husband's great granfather. He was born in 1885 with no father listed and there is a father on the marriage cert. In my case, the father's name is identical to the illegitimate child - easier to remember if questioned??
Anyway, one thing I am trying, that you can also try, is tracing the people he's lived with. His mother died when he was 12 (1897), so I am tracing the family he's living with in 1901. Maybe I will find something?

Even if I do find something, it will be a guess and I probably won't know it if I stumble over it.
I am thinking that DNA testing may be the only option. Luckily for me, it's my husband's paternal line.
Does your person have siblings? Did the mother eventually marry? Supposedly, it wasn't uncommon for the mother to marry the father at a later date. If you have siblings, you could try tracing their father, as that's a possibility. For me, mine was an only child - another dead end.
I would love to hear other ideas.