I mean one of my recent TLs explored a Federal EU in depth which had a multiparty system traditionally dominated by centre-right Christian Democrats with Liberal and Social Democratic opposition that was now swinging towards extremism on both sides. There was a kinda standard, bicameral parliamentary system with a "Minister-President" as leader. Governments were always inevitably
I solved the proportional/representing regions issue but having each state as a constituency with a certain amount of seats which then allocated proportionately for that country. (say France has 100 seats, if 50% of French votes went to the blue party then 50 seats would go blue.) That way you have a broadly representative system but small states still have representation of their views and, for instance, German votes and French votes will always be counted separately.
If your EU is anything like OTLs it'll probably be fairly socially progressive but with a large range of views, I represented that with 7 parties ranging from Soviet puppets and Green Trots on the left to Christian Nationalists and post-Fascists on the right.
Honestly it really depends how you want to handle it but its going to be multiparty, almost certainly be parliamentary in nature and probably remain very decentralised.