Oh dear! Desperation time, I think. Tell me more about this unwritten Constitution.
well first of all it"s a common misconception that the constitution is unwritten as a whole. There are many written pieces of it made up of multiple pieces of legislation such as the Bill of Rights, the Settlement Act, The Acts of Union,and indeed the European Community Act which will need to be repealed at some point in the lead up to Brexit.
It's also a misconception to think that those parts of it not codified into legislation are some how non existent, as while they may be difficult to enforce in a court of law, would present extreme difficulties in not following. An example of this would be the Royal Prerogative, which while it now has a set of guidelines, is not part of legislation.
In addition there are of course a number of common law principles which while not part of legislation cover some parts of a constitution such as the case covering the rule that anything not prihibitrd specifucally is allowed.
Overall in this case, the idea that a Prime Minister has the necessary power to agree this sort of decision unilaterally, is negated by both the method of government control and feasibility given the primus inter pares approach we have.