It's actually a bit ingenuous to call EU citizens "immigrant labour"
All EU citizens have the right to live and work anywhere in the EU (apart from some transitional arrangements, which the UK government perhaps unwisely didn't choose to implement).
As I have, the right to live and work in Spain (though I'm retired now). This doesn't mean I would have to work for the minimum wage though - as a person with a skill I could get a skilled job at a skilled workers wage. So your Even Davis example just wouldn't occur in the way you suggest.
And you must also remember that lower wages can lead to lower cost of production, and so to lower prices. How would it be, for instance, if food prices shot up because productivity was lower and/or wages were higher? - howls of protest I would assume.
Or if the cost of manufactured goods went up, more cheaper imports came in, and so jobs were lost from the UK completely? That scenario has already decimated UK manufacturing.
One issue is that for seasonal workers in particular, the benefits system makes it very difficult for them to do just a couple of months seasonal work. Just another example of a lack of flexibility in employment law actually works against people in some circumstances.