from my perspective, the Leviathan appears to be more a state-building theory rather than a theory of international state behaviour.
Hobbes is a precursor to the classical realists
The Stanford Enyclopedia of Philosophy is an excellent resource for researching political theory. This answer is based on their entry on political realism in international relations.
Hobbes' theory is not a part of classical realism, but is a pre-cursor to it. The key difference is that he is mostly focused on domestic, rather than international politics. His theory does not explicitly capture the kinds of international behavior that the classical realists do.
Domestic vs International Relations
You correctly pointed out that Hobbes is largely focused on the relationships between individuals and the sovereign. He does not explicitly deal with international relations. Although (in my experience) many readers apply the state of nature to international affairs and therefore imagine that states behave like individuals, this is not a part of Hobbes argument.
The classical realists are more explicitly focused on international relations. They construct a world populated by state actors who pursue their own national interests. There is no effort to say that states are "like" people. Rather, state actors have their own interests which may (or may not) reflect their citizens' interests.
Hobbes does not suggest any kind of international "order". While individuals can escape the state of nature by agreeing to a sovereign to impose law, Hobbes does not propose a similar mechanism for states to ensure their continued security. The key difference seems to be that individuals institute a sovereign for their own security, but international insecurity for the state does not translate to individual insecurity. Therefore, there is no reason to institute an international "sovereign".
Conversely, the classical realists see international relations as a struggle for power, not security. States may come to agreements which enhance the power of all parties involved, as long as the agreements remain in effect. Realists therefore allow for international "order" in a way that Hobbes did not attempt to.
SEP includes one other interesting dimension. Hobbes' state is largely defensive. It acts to secure the security of its citizens. The classical realists' focus on power turns international politics requires an offensive strategy.