Dear OP
Different perspective here. If DP is prepared for a potential court fight and to deal with the emotional effects for all involved, then maybe he should pursue it now.
Because this boy will find out at some stage. Either a DNA test, his mother finally tells him eventually, your DP tells him when he's older, another family member lets it slip one day (and it won't be a secret, people will know - maybe her parents, her siblings, her aunts and uncles, cousin, family secrets a really that secret). Or even his current 'father' might one day blurt it out because he is facing a stroppy teenager that isn't actually his. And I know someone that happened to - it was traumatic.
If your DP can get legal access whilst the child is young and can build a relationship slowly over time, in the long run that may be much better than the shock of finding out as an adult, though obviously not easy.
Three situations from my wider family to illustrate.
2 are the child in this situation. One found out at 17 he was not his father's biological son after a family friend said something when drunk at a family gathering. Both (non-bio) father & son were devastated and it caused a rift that took a long time to heal as son found it hard to forgive the years of lying.
Another was adopted but before that had been brought up to think Man A was her father. Was told at 6 by her adoptive parents that actually Man B was her bio Dad but he had left before she was born Took it well, no contact with bio-Dad asa child and when she went to find him as an adult, found out he'd died. So no chance to ever get to know him.
My brother in law was your DP, although he always knew he had a son but the mother persuaded him not to be involved as she had a new boyfriend by the time child was born. Child was told at 18 but chose not to have any contact BiL as they assumed hadn't been wanted and felt rejected. They finally met when child was in 20's, BiL apologised for not having fought harder for access as a child and both now deeply regret it, especially as child had a difficult, neglectful childhood with mum and the boyfriend.
OP, There are no easy answers here. If DP does decide not to go ahead now, then the suggestions of keeping cards, writing letters and putting some money aside for when child is older are good. It's going to be important to be able to show he cared so son doesn't feel DP rejected him once he found out.