I don't really see how using a haploid affects things either. Mary's DNA provided the human part; unless, the concern is that there would be no Y chromasome coming from Mary, being female. Of course, considering that this is a miraculous event, providing Jesus a Y Chromasome would be no problem if making him a male human is an issue. The Holy Spirit did not place a baby in Mary's womb. The Holy Spirit created a baby through Mary.
Now, as to using Mary, God could have incarnated himself without human agency at all. However, He wouldn't have because the purpose for using Mary is to fulfill his promises to David in Scripture. The Messiah had to be a blood descendent of David. I think that is what you are getting at with the lineage question isn't it? Or have I misunderstood what the debate is about?
His blood descent of David is through His Mother and that's not how the Law sees it. His descend of David through His adoptive male parent, St. Joseph, is one of adoption and not blood lineage.
However, more importantly, having been incarnated of a Jewish woman, He is related to Abraham in flesh, and to all humanity in nature.
Yes, that is exactly what I'm talking about. We're on the same page.