None. Very few things are managed on a federal level. It's much more akin to Austria-Hungary than the US. The largest federal entity would be the common war ministry headquartered in Tbilisi (strictly due to its strategic and central location) which manages the union's armed forces; Georgia being responsible for the navy, Azerbaijan with the airforce, and Armenia and Dagestan sharing the responsibility for the army.
Does it have an equivalent to Brussels in the EU? Speaking of the EU, it strikes me calling it a 'federal republic' is an unlikely choice of name: 'Caucasian federation' or 'Caucasian confederation' or perhaps even 'Union of Caucasian Republics' seems more fitting, and less likely to ruffle nationalist feathers. It's not like it's a state in which which '
supreme power is held by the people and their elected representatives' , rather it's an organization in which supreme power doesn't really exist save where it's negotiated between the elected representatives of several different peoples. (When you say Georgia is responsible for the navy, does that mean the Georgians pay the taxes that support the navy? Who gets recruited into the navy - only Georgians get conscripted, but maybe other nationalities can join voluntarily?)
(btw, does anyone know how were taxes to pay for the military raised and soldiers recruited in post-1867 Austria-Hungary? The articles a quick google throw up say " the army, foreign affairs, and related budgetary matters remained with
the emperor", but that's a bit short on specifics.)