The problem arises because the Government would usually use the Royal Prerogative for this.

So what's at stake is the balance of power between Monarch and Parliament.

If the Prerogative doesn't work for this, what else doesn't it work for?
How? What circumstances are you thinking of, if we have already covered those in which the Parliament explicitly invokes the article and the government proceeds with negotiation, the Parliament rules out the referendum result and maintains the status quo, or the Parliament rules out the referendum with the caveat that the government is directed to otherwise seek modifications or accommodations to EU membership standards (as the UK has always done)?

What specifically could happen or come into legal conflict in these or unmentioned circumstances? Clandestine diplomacy at the highest levels? That would be a more run-of-the-mill political crisis.

Edit: There's also the basic principle that, explicit or not, the referendum was intended to settle the question - now Parliament might settle the question.

So you actually have two things at stake - executive power and the value of the popular vote.
Are you sure you haven't gotten confused and weren't just thinking of this the whole time?