The logical flaw in the Brexit argument (you can't have your cake and eat it)

Martin Lewis, of Money Saving Expert fame, has published this careful assessment of the pros and cons of Brexit and there is little I can add to it, except to point out that it outlines neatly what I see as a fundamental logical inconsistency in the Brexit case (without naming it as such).

Briefly, the strongest Brexit arguments, as I see it, are these (I honestly think that the democratic deficit argument is of real interest only to politicians and the economic upside/downside argument has left us all baffled):
either
1) Leaving the EU will protect us from the invasion of immigrants who are taking our jobs, lowering wages, swamping services etc. This implies that the alternative to the EU is an economy that is properly regulated by the UK government, with less immigration and more guarantees of jobs, housing and services (health, social care etc) for UK residents.
or 
2) Leaving the EU will free the UK economy up from burdensome regulation on issues such as workers' rights and enable us to make free trade agreements with economies across the world. This implies that the alternative to the EU will be, in Martin Lewis' formulation, "a nimble low-tax, low-regulation, tiger economy. Trading unfettered with all nations across the globe, able to create our own rules and speedily reacting as a niche player to a changing world."

The thing is, I have presented these advantages as either/or because they really are mutually exclusive. A "nimble low-tax, low-regulation, tiger economy" won't limit immigration for long if immigration provides for cheaper labour, particularly if it wants to "[trade] unfettered with all nations across the globe." What's more, the "rules" such an economy will create will absolutely not be rules that hamper business growth by protecting British workers' rights. And "tiger economies" are not generally known for their social protection. No place for an NHS in a tiger economy, not with all those rapacious entrepreneurial multinational healthcare firms out there.

So what I'm saying is, if either of the two arguments above is on its own sufficient to trump the Remain arguments about peace and stability, communitarianism rather than isolationism and the desire to make common cause with our European allies, and on its own sufficient to justify the leap in the dark that Brexit will inevitably involve AND you believe that you will get to choose the version of standalone Britain you like then go ahead, vote leave.

But don't go thinking you can use both justifications simultaneously.