I don't have any designs for you, but since this seems to mostly revolve around a unification of Germany and Italy,
The Italo-German duo is obviously the hegemon of the union, but it shall also include all the other states of the CP bloc (H-C-S, Greece, Egypt, and Scandinavia) as founding members. France and Iberia shall join it when the CP let them out of the doghouse. As it concerns the CP Europe-Russia relationship, well, things may still evolve in several different ways over the next century. As it concerns Britain and Ireland, it seems like the story calls for butterflies taking a different path than joining Europe.
and since they're de facto sticking to a monarchical system, how about just the Holy Roman Empire, or Holy Rome or just Rome for short?
The CP states are in alll likelihood going to stick to a constitutional monarchy system for the foreseeable future, no plausible good reason for things to go otherwise. France and Iberia instead may easily become and stay republics after the postwar political chaos subsides, given how much the Bonaparte and Bourbon monarchical experiments massively screwed them again and again. About Britain, the coming total defeat shall surely bring them to a revolutionary situation, but it could evolve in various different ways as it concerns the fate of the monarchy. It does not seem that even in the plausible worst-case scenario, things are going to be so bad for Russia that they would face a revolutionary situation similar to OTL, although they may certainly be some serious political backlash if they lose the war.
TTL CP Europe is certainly using a mix of Roman, Carolingian, and HRE heritage as the ideological basis of its unification.
I gave some thought on how its successful monarchical tradition would influence the institutional architecture of the *EU/EF in comparison to OTL. I have tentatively come to the conclusion that in all likelihood, the main differences, if any, may involve the European head of state figure.
The legislative, executive, and judiciary branches would not really be that much different from the parliamentary system of our modern EU. There would still be a directly-elected Legislative Lower Chamber, an Upper Chamber made up of the representatives of the various states, an Executive responsible to the Parliament, a Court of Justice, and a Bank.
A restoration of the HRE with a figurehead European Emperor is a distinct possibility with a lot of cool and tradition, but it has two difficulties: first, several CP states are already Empires, and in the European tradition there is no precedent for a dignity superior to the Imperial one, it would have to be invented out of whole cloth; second, the European hegemon is a diarchy, which makes the creation of a single super-Imperial figure a bit troublesome.
Other plausible possibilites include a rotating or collective presidency of the various head of states, which I think may have greater chances of adoption given the situation.