Interesting topic.

If the state is defined as the only legal perpetrators of violence…

Is that the right definition? It could be a good place to start. You might want to look up John Austin. This article may be useful: https://plato.stanford.edu/entries/legal-positivism/ If Austin is useful, you might want to check exactly how he reflected Bentham and then see if Marx said anything about that part of Bentham. Marx wasn't generally kind to him lol.

If you find this legal view of the state and keep reading you may come across Kelsen. If so, or if you're otherwise interested, have a look at Max Adler, The Marxist Conception of the State: A Contribution to the Differentiation of the Sociological and the Juristic Method. It's an old text but may be useful.

Otherwise, is the state the 'only legal [perpetrator] of violence' or does it have a monopoly on violence? Is there a difference?

I've yet to read it but Ralph Miliband's The State in Capitalist Society could be useful, too.

If you are going down a legal path, Pashukanis and Renner may be essential reading. Even anti-communists treat them seriously. I'm unsure if he's an anti-communist, but Fuller's Morality of Law relies on Pashukanis. Fuller was involved in one of a few great debates in jurisprudence in the twentieth century. His was the Hart-Fuller debate. Hart being the arch-positivist. You'll see him mentioned in the Austin article cited above, along with Kelsen (Pashukanis gets a line about some Marxists rejecting positivism).

(For those who are just passing by, legal positivism is basically the idea that there is a difference between what law is and what law ought to be. I.e. law doesn't necessarily tell you what is moral and law doesn't necessarily have to be moral.)