How do we know that the 1-4% shared DNA between humans today and neanderthals is from inter-species procreation?

They ruled out convergent evolution because that's not how it works.

They ruled out an earlier ancestor because its only some humans that have Neanderthal DNA, Africans don't have such DNA (other than those who have non-African ancestors), which is very good evidence that the interbreeding took place after modern humans started migrating from Africa (as Neanderthals did not live in Africa).

I know there is a small group of scientists that believe the DNA in question could have come from an ancient ancestor. The problem is that modern humans and Neanderthals split from a common ancestor probably around half a million years ago. To believe that such old DNA sequences could be found in both species after so much time is not really believable.

The term "inter-species" is a misnomer. If indeed neanderthals and homo sapiens were interbreeding and producing viable offspring, that means they were the same species. It's the same thing as saying black people and white people are different species. Clearly that isn't true.