I give this type of argument the name "moral presuppositionalism".
The claim here is that we can axiomatically define morality to include carnism, and therefore carnism is justified. Another form of the argument is "I will presuppose category X is justified. Carnism is of category X. Therefore carnism is justified." This fundamentally underpins every single argument of the designation EH, AA, and AH on this website, amongst others e.g.
And infinitely more...
Nothing prevents me from axiomatically defining myself as the only being towards which moral obligations exist. Murder is now ethical.
Unless it focuses on the reduction of needless harm toward the well-being of conscious beings, it can't be morality. Even if one axiomatically defines indiscriminate murder as moral and anything else immoral, is that really morality? Or if someone takes a less blatantly silly position and says "it's okay to harm certain animals in certain ways". You need to provide a justification or it's just the fallacy special pleading. It's an argument that's indistinguishable from something you pulled out of nowhere for convenience.
Imagine you and a friend are walking down the street when you see a shiny bright blue Porsche and your friend turns to you and says "You know, you and I have identical moral values concerning stealing in any other context... but I will assert that blue cars are okay to steal unless you can prove to me that it's immoral". It takes very little to see how poorly reasoned such an argument is because it shifts the burden of proof. It practically cries out for a response akin to "where did you get that oddly specific and highly convenient rule from?!". But substitute in "it's okay to harm the well-being of specific animals in specific ways unless you can demonstrate that it's immoral" and all of the sudden it's impossible for some people to identify why this argument is silly.
The problem is just asserting that it's okay to harm specific animals in specific ways is an unfalsifiable assertion. It's a distinction that is made with no backing or justification. Hence you don't just get to assert it and then ask the other person to prove otherwise.
If we want internally consistent morality, we have to take into account the human impact of eating animals. If one wants to then axiomatically make it ok to hurt other people in all the ways that carnism does, it starts to become ok and not ok to do highly arbitrary combinations of things.
Eating animal products inflicts needless suffering and reduction on the well-being of conscious creatures for one's own entertainment, both animal and human. There's no good justification for this in any reasonable moral system.