A Google search suggests this constitutes procedural impropriety
IMO, Google is wrong. Procedural impropriety is not statutory grounds under the RTA. But if you were so minded you could include this within the scope of 'penalty exceeded .....circumstances of the case'..which, as you can see, has been stated incorrectly anyway!
But IMO, there's nowhere to go with these points at this stage, you need to wait for the NTO. So, are you the registered keeper with current DVLA details?
Talking of the NTO, IMO let's hope they serve this as soon as practical after the 14-day discount period has elapsed. This is because you are in a very rare situation i.e. your initial reps and their reply occurred so soon after the PCN was served, that the authority MAY NOT serve a NTO as stated, they have to wait until 28 days have elapsed: 20 March plus 14 days is 13 days short of the 28-day period.
..
..
1)Where—
(a)a penalty charge notice has been issued with respect to a vehicle under section 66 of this Act; and
(b)the period of 28 days for payment of the penalty charge has expired without that charge being paid,
the ..
authority concerned may serve a notice (“a notice to owner”)