"I suppose the problem they have is that going to modern scholarship turns up lots of inconvenient issues which would need to be airbrushed out."
Sort of true. Whilst there would be no problem if they wanted to use modern scholarship to explain Josephus' and Tacitus' references to Jesus, they would have to deal with certain inaccuracies in the Bible if they wanted to use the Bible itself to "prove" Jesus' existence. For those who have no or little knowledge of how history works, the Bible can be used to explain Jesus' existence. Let me explain.
For example, one of the evidences that Jesus did exist is the fact that Jesus dies in the gospels. The problem is that in this period of time, Jews did not believe that the Messiah would die. They had many different ideas about who the Messiah would be and what he would do; however, there was no notion that he would die. "Messiah is sent by God, and then dies—on a cross, at that? How the heck does that work!?" would a Jew of that time answer. The belief that the Messiah would die developed after Jesus' death to explain why he died when he was the Messiah.
Now, if the Watchtower wanted to use the Bible to "prove" Jesus' existence, they would have to refer to such explanations that would be inconsistent with their own beliefs. In their publications, they have numerously claimed that the Jews expected the Messiah to be the priest, the king, and the prophet all at the same time. I believe there was a study edition of Watchtower last year which explained this concept. This is a completely inaccurate assumption.