The letter writer isn't specific enough, but it could be that mom blocked dad's attempt to have more official time with his son and this limited contact isn't his idea.True but picking up another child who is in the same class, right in front of his own son, was apparently his idea and it is extremely insensitive. I feel very sorry for his son.
But what is the dad supposed to do? Tell his girlfriend that he can't pick the child that lives in their shared home up from school because his son might see him? And there is absolutely no indication that the dad doesn't take that opportunity to say hello and give his son a hug. All we have is the view of a not-so-disinterested 3rd party who is assuming that Boy #1 is acting up because Dad is picking up another child from school.
Of the top of my head as alternatives: 1) per the OP, the child is going back to the house where the girlfriend lived before the ex-husband moved in, so she must have had a previous arrangement to get the child home. Use it.
2) Try buses or whatever public transport is available.
3) Tell the girlfriend to pick up her own child to avoid upsetting his son.
Even if he does give a quick hug, it doesn't make it better when it's still another a child the father is taking home, one from his own class no less. It might actually make it worse. As the two adults that did the damage, the onus is on the father and his girlfriend to do what is needed in mitigation.
(Although I'd suspect that it's not so much the pick up after school causing the problem as other children in it. "Traded in for X" is a fairly common taunt of playground bullies, whether or not the "new" eight year old knows or takes part. There simply isn't enough information to confirm that.)