Here was my approach. I identified that “since” is support. That means the blank must be a synonym for objection, I thought of “cavils with.” I guess I got tripped up with the second blank: I thought that the “but also” should be a continuation of the negative idea that his objections would be irrelevant and something like “baseless.”
Where did I go wrong? If irrelevant is the clue for the second blank, but the “but” acts as a contrast, then the blank should be opposite? Let me know what the GregMat hive mind thinks.
Hey, what makes something irrelevant? If the author actually row with scholars and give objection on fighting, then that’s irrelevant. See it like cause-effect relationship, author frequently does ‘sth’ to other scholars that has two effects i.e irrelevant and ‘sth’.
Since the first blank is filled with ‘row with’, the second can be filled with surprising because that’s the effect of something that’s happening there.
It’s not that ‘but’ acts as contrast but it is a continuation itself.
Hope you get the idea 
Right, I interpreted “but also” as a continuation, so I thought the second blank should go along with irrelevant. If something is irrelevant, it’s immaterial. Where are we getting surprising?
Also, I’m not sure what you “sth” is. Why would it be surprising to people if everyone understands this guy always cavils or objects other scholars?
Here the guy always cavils is correct but he does not objects other scholars, rather he objects the dispute. I think this is where you misunderstood.
At one instance, he fights with the scholars and the other hand, he objects to fighting. We need to select a word that solves this paradox, irrelevant and surprising are the only words that fits. We can rephrase it like: It is surprising that author frequently quarrels at one instance and rejects fight at the other instance.
So he is kind of a hypocrite for caviling at other scholars, but then objecting at fights. Ok.