Where Christ used Psalm 110, and where he didn't

I'm on chapter 12 now. I got a bit sidetracked on account of today being cloudy, occasionally rainy, Sunday, and hot. So couldn't leave the house, so ended up napping 10 AM - 12:30 PM.

On Pitre's assumption that Mark is Peter's memoir (which assumption I share) the Christian exegesis of Psalm 110 is mooted. I remember this from [pseudo-]Barnabas. Also from Anon To The Hebrews which Pitre concedes is nonPauline so might be late, perhaps even Barnabas-contemporary; in fact Tertullian ascribed Hebrews to exactly Barnabas (being unaware of the other one).

The orthodox Christian reading of Psalm 110 (and of Melchizedek) is that - having accepted this Psalm - it proves the existence of a Messiah who is not the son of David, or at least not only that. Christians in the Hebrews / Barnabas generation, which was likely post-9-Av, were fond of this one.

My concern, against Pitre, is that Barnabas was aware of a Jesus biography. Might have been oral; might have been written - I don't know. Mark's gospel is possible; that Christ shall be seen at God's right hand is a Psalm 110 flourish which Pitre #11 sets in Mark 14:55-64.

What is not possible, for Barnabas, nor for that matter for Hebrews, is that Jesus cited this Psalm for his own Christology. Barnabas like Hebrews and like Mark himself nowhere ascribed this particular reading of the Psalm to Christ. These three were all aware that Christ had come to Earth, done deeds, and said sayings; Mark wrote a whole book about all that. They were also all aware of Psalm 110's relevance. Why not this bit of Christology, a tafsir bi'l-masihi if you will?

Cerinthus objected to higher Christologies on adoptionist terms, hence why his sect held to Mark to the other evangelists' exclusion. But even Cerinthus was stuck with Mark's pronouncements of some part of Christ's divinity. If Cerinthus had been aware of (say) Barnabas it is unlikely he'd have objected.

Mark may or may not have been aware of Barnabas, or of Hebrews. I find plausible Mark agreed with that exegesis of Psalm 110 now seen Mt 22:41-6/Lk 20:41-4. But I find equally plausible that nobody was aware that Christ himself had cited the argument. Until Matthew stuck it there.

Pitre's book is full of stuff like this, on assumption of Matthew as an eyewitness (if secondary to Peter); to bolster the highest Christology. Pitre does this even where he doesn't need to. By doing this, Pitre is undercutting his own credibility and, thereby, the credibility of the Gospel. I find this to be unnecessary and, honestly, tragic.