The “very specific narrow duty” reading is not a plausible reading, if it suggests that the Government could send simultaneous messages (public or private) contradicting the required letter.
I can’t see that any court would read the Act that way, as it would frustrate its self evident purpose. Why would Parliament have bothered to require the Government to write a letter if the Government could simultaneously subvert it?
Use of a power to “thwart or run counter to the objectives” of an Act of Parliament is a well-established ground of judicial review: the Padfield case (1968).
AKA: cunning plans based on a narrow barrack-room lawyer textual reading of a statute to get round the obvious purpose of the statute don’t work.
The Court* would have power to grant an order prohibiting the Govt from making any such contradictory communications; and to order it to rescind any such communications (making it clear they were not the U.K. position).
*in England and Wales: Scots courts may have different or additional powers.
As to the EU, as a body governed by the rule of law, and given the reference to the departing state’s “constitutional requirements” in Article 50, it seems to me that the EU would have to take the departing state’s position as being that which its own law required it to take.
If it received two letters from the PM, letter 1 in the required form under the Act, and letter 2 saying “please ignore letter 1: refuse an extension” then if a U.K. court had ruled that letter 2 was not a lawful exercise of the PM’s powers, the EU would have to ignore letter 2.
Letter 2 would be of no more legal effect in law (EU or UK) than a letter written to the EU Council by a Mr ABD Johnson of Westminster.
As to a letter 3 saying “I have been required to write letter 1 but I give you notice that I intend to make no use of any extension period as I’ve no intention of negotiating any further”, that letter would not affect the UK’s legal position (it would be requesting an extension).
It would be up to the EU how to approach letter 3. But there is nothing stopping it from noting the position of the Johnson Govt in the HoC, & the position of the parties on a general election, concluding that a GE is almost inevitable, and granting an extension for that reason.
