The first one, though it sounds good, is ungrammatical because the infinitive perfect is being used next to an already stated past tense. An infinitive perfect should be used with a present tense, to express an infinitive in the past. Otherwise it should just be a normal infinitive.
In the example, "He became (simple past tense) the first 16 year old", a simple infinitive is necessary to qualify the same time frame as when he became what he was. It was an instant moment, that he became what he was, and that he scored the goal.
Here are a couple of examples of when you would use the perfect infinitive. You will notice they are used next to a present tense to show a distinction between past and present.
"It is necessary to have scored a goal for your country before the age of 16, in order to qualify for this award."
"It is better to have loved and lost, than never to have loved at all."
And here are some alternatives to the example in the original question that should qualify to use the perfect infinitive:
"I am the first 16 year old to have scored a goal for my country." – (it happened in the past, I am speaking in the present)
"He was the only 16 year old I knew to have scored a goal for his country." – (it happened before the event being told)