I'm not really sure how you tie the immigrant issue into an idea of corporate oligarchy running the EU. If by immigrants you mean the refugee crisis, then I'm not sure the corporate oligarchy would have a unified view on the issue. ...
Two things: first I lump refugee with wilful immigrants -- both are looking to improve their lot -- and second I think immigration should be without barrier or constraint, certainly without being penned up in counterproductive camps. If more people were able to vote with their feet perhaps wars would not be so prevalent.
But they would also tend to go where the benefits and pay are better, so companies would have to up their share to the workers to keep workers. Corporations tend to like keeping a cadre of cheap(er) labor.
Worth bearing in mind, however, is the enormous day-today benefits we gain from the unification of the European market ...
Going to a common currency certainly helped unify the various states in the early US history, so a common currency is not a bad idea, but does it need a whole government structure to implement? Would not the world benefit from a single currency if that were the case? Probably.
... and the abolition of trade barriers ...
The abolition of trade barriers between states was also a good thing in the early US history, but the elimination of trade barriers between countries (NAFTA etc) with significant differences in the way workers are treated has been a disaster for the middle class workers in the US as jobs are shipped overseas for cheaper labor and lower safety and environmental controls.
Again, open borders can accomplish the same thing without another layer of government oversight. and the tendency of corporations to go where low wage workers are would be countered by the tendency of workers to go where the better working conditions are -- if they had the freedom to go.
Personally I think the CEO and Board of Directors model of corporations needs to be changed to a democratic worker owned company model.
Google Basque Worker Co-op for some interesting information or see
Mondragon Corporation - Wikipedia
Enjoy