Fraa Lodoghir is far too canny an operator to baldly tip his hand, but consider the overall plight of the avout relative to the Saecular Power. Consider also the deal Tulia springs on Erasmas at Eliger - she will join the Procians in the New Circle, to try to build protections for the Edharian faction at the hierarchical level.
Recall that she more than once points out to Ras that he's lousy at politics. Orolo also claims to be notoriously bad at politicking. Likewise, before we meet Lodoghir, we're told the upcoming Plenary is all theater.
Lodoghir is FAE of the Centenarian chapter of the Order of Saunt Proc at Muncoster, one of the hyper-influential "Big Three" concents, and also - critically - the one where the Parking Ramp Dinosaur occurred. Recall that as Ras explains matters, Procians began that experiment, while Halikaarnians ended it. Recall as well that it was that incident which triggered the Third Sack.
Convox, by definition, is called by the Saecular Power, and brings the avout there under their scrutiny.
Lodoghir takes an adversarial stance toward Ras in Plenary so that it looks, to the Powers That Be in attendance, as though Ras is getting a Mathic dressing down. The Saecular Power was provoked by the Visitation - Orolo or the Edharians at Ecba were not, in its view, authorized to make contact with the Geometers.
But Ras is a lousy actor. Safer for Lodoghir to take an adversarial stance and allow his reactions to be genuine.
Similarly, in the messal. The Saecular Power didn't trust Ignetha Foral to have alone-time with her pet Cosmologist-Metatheorician, Paphlagon. Emman Beldo, nominally her chauffeur and servitor, was doubtless her minder, and Saecular eyes on the messsal.
Lodoghir's dismissive remarks were said TO Erasmas, but they were said FOR the Powers That Be. In messal, he plays his self-assigned role of Sphenic gadfly - a.) It's his cover but b.) In Dialog, it's a CRITICALLY USEFUL role in developing an argument.
This is why, after events, he is solicitous of Erasmas and others of the Cold Mirror team. One, he's making sure they don't say anything honest but impolitic. But also, he's not actually the Sphenic beast he's pretended to be. But he only ADMITS to Procian/Halikaarnian cooperation after the fact. I feel it's been his modus operandi all along.