I think it might be more complex than that. I would expect that most of the upper sections of the native working class and new middle class will gain from immigration via cheeper prices and services while loosing very little in terms of wage competition.
At the same time those who are directly competing with immigrant labour are likely to be unskilled workers many of whom will be immigrants themselves.
What I would expect is that there will come a breaking point, like there did in the US a while ago, and there will be a big drive by immigrants to organise themselves, but this will not be on the basis of calling for open borders and further European integration - possibly the reverse!
At the same time there will be greater stratification and inequality within the working class making its overall position weaker.