As you must know by now, you are reading a book in which the characters are constantly using allusive language and expressing subtle humor through irony--to say nothing of 1940s slang!
A "cold cut" means cold, thinly sliced meat--for example, thin slices of ham for a ham sandwich.
Someone whose personality is distant and emotionally cold is sometimes called a "cold fish."
I'm certain the speaker is criticizing Dewey as emotionally cold, or lacking in empathy, or does not communicate enthusiasm. Wikipedia notes that that many "criticized his perceived stiffness, coldness, and aggressiveness in public. One of his biographers wrote that he had "a personality that attracted contempt and adulation in equal proportion." An opinion piece in Slate said in 2008 that the Republican strategy was to "Deweyize Obama," i.e. depict Obama to be a cold, distant snob.
Why he is using the specific phrase "cold cut," I'm not sure, but we don't need to know. Obviously it's insulting to compare Dewey to cold meat.
I'm now going to make a wild guess.
A popular kind of cold cut in the U.S. is sliced baloney--a kind of mild sausage, once spelled "Bologna." "Baloney" is also a euphemism for the vulgarity "bullshit." If someone thinks an authority figure is using clever language to express something disagreeable, they are likely to refer to the saying "No matter how you slice it, it's still the same old baloney."
It is possible that the use of the phrase "cold cuts" is supposed to make us think of "baloney." Maybe, maybe not.
Cultural note: Dewey ran against Truman. It was such a close race that, based on incomplete election returns, some newspapers actually reported that Dewey had won. There is an extremely famous photograph of a triumphant Harry S. Truman grinning as he hold up a newspaper with the headline "Dewey Defeats Truman!"