If he is not born in Qatar, but he acknowledges on the birth certificate that he is the father, then i don't see any reason why the child can't be under the father's sponsorship? Illegitimacy is not entertained, that's a fact. But where in the birth certificate says that the child is illegitimate?

I don't see any problem. If the child is born in Qatar, then, that's another problem. Proceed with the application. It wouldn't hurt your BF to apply. Declare that he is applying for his child's sponsorship.