I don't see this as a problem. It's something that can be agreed to disagree upon. I like Carrier's use of Bayes' Theorem there, since I think that may have something to add to the topic.
I agree that Second Century literature may help. But why say we can use 2nd and 3rd century Christian literature on the question of historicity, and at the same time rule out the NT? It seems there is an inconsistency there. Can you explain this?
Theophilus' Apology to Autolycus, written around 180 CE, does seem to refer to Jesus Christ, though the name "Jesus" is not used. From here:
And hence the holy writings teach us, and all the spirit-bearing [inspired] men, one of whom, John, says, "In the beginning was the Word, and the Word was with God," showing that at first God was alone, and the Word in Him. Then he says, "The Word was God; all things came into existence through Him; and apart from Him not one thing came into existence." The Word, then, being God, and being naturally produced from God, whenever the Father of the universe wills, He sends Him to any place; and He, coming, is both heard and seen, being sent by Him, and is found in a place.
What is your interpretation of that passage, if it isn't a reference to Jesus Christ?
And Papias certainly refers to Jesus Christ, as reported by Irenaeus and Eusebius, though again the name of "Jesus" is not used. From the Fragments of Papias:
http://www.earlychristianwritings.com/text/papias.html
But I shall not be unwilling to put down, along with my interpretations, whatsoever instructions I received with care at any time from the elders, and stored up with care in my memory... If, then, any one who had attended on the elders came, I asked minutely after their sayings,--what Andrew or Peter said, or what was said by Philip, or by Thomas, or by James, or by John, or by Matthew, or by any other of the Lord's disciples...
The "Lord" in this context can only be Jesus Christ. Not only that, Papias claims to have met people who knew Jesus' disciples!
Similarly, Irenaeus reports that Polycarp, a contemporary of Papias, also met people who knew Jesus' apostles. From here:
http://www.earlychristianwritings.com/polycarp.html
But Polycarp also was not only instructed by apostles, and conversed with many who had seen Christ, but was also, by apostles in Asia, appointed bishop of the Church in Smyrna, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man, gloriously and most nobly suffering martyrdom, departed this life, having always taught the things which he had learned from the apostles...
For, while I was yet a boy, I saw thee in Lower Asia with Polycarp... I can even describe the place where the blessed Polycarp used to sit and discourse-his going out... Whatsoever things he had heard from them [apostles] respecting the Lord, both with regard to His miracles and His teaching, Polycarp having thus received [information] from the eye-witnesses of the Word of life, would recount them all in harmony with the Scriptures.
Irenaeus was born early in the Second Century, and as a boy he actually meet (or claimed to meet) Polycarp, who was born in the 60s CE. So from a timing perspective, Polycarp may well have met apostles who had met Jesus.
Irenaeus claims to have met Polycarp, whom claims to have met some of the apostles who met Jesus. Papias claims to have met elders whom knew the apostles who met Jesus. These are interesting claims.