And if Anti-Dühring is too much for him he can just read through Socialism: Utopian and Scientific, an excerpt of the former which like the Manifesto was published as a polemic meant for a large audience that has not previously read much about the topic.The Communist Manifesto is a short propaganda pamphlet designed to get people excited about the possibilities of communism. It's an interesting historical curiosity in the vast body of Marx's copius output. If Peterson really wanted to engage with Marx's thought, but not slog through 800 pages of Das Kapital, vol 1, he should have read The Eighteenth Brumaire of Louis Napoleon, which is about historical analysis and which is were we get the phrase "First as tragedy, then as farce", or Friedrich Engels' Anti-Dühring, which is a basic survey of how communism is supposed to come about, function, and how it differs from socialism. Both are books one could finish easily in a weekend, and come away with a very clear understanding of the sort of thinking that Marx and Engels were writing about.
Though it was hilarious listening to Peterson basically admit that he couldn't understand Zizek's writing, and had to resort to the one book on communism that everyone's heard of.
Also interesting in my view (though not necessarily entirely relevant to his theories on socialism and communism) are Marx's frequent contributions to the New York Daily Tribune. They give you a little background into the politics of the day and a sense of what political context Marx was writing in. Some of them are available easily with Marx and Engels' other writings on marxists.org, including a couple articles by Zizek himself on Lenin. It shouldn't take too much digging for Peterson or any other enterprising theorist looking to analyze the writings of Marx and Engels or almost any prominent Marxist or otherwise left-wing writer to find what they need. It's one thing for Peterson to come into a debate on Marxism with little more than a normie-level understanding of what Marx was on about, but another thing entirely to come into said debate without even glancing at his interlocutor's work, with several months' time for preparation.