Whether or not it is de minimis is not up to the council or your builder/architect to decide because a height limit is a height limit and the line has to be drawn, hence the reason the council must refuse a LDC; an application for one would be a waste of time and money.
The question of de minimis is only raised by a planning inspector adjudicating an appeal (if, for example, the council refused a Planning Application) or - ultimately - by a court.
There are guidelines issued by central government to councils on enforcement policy. One part states: "enforcement may not be appropriate for a minor or technical breach where no harm has ocurred". Maybe your situation falls within this - the only way you would know is to go through the Planning- and appeal process - ughhh...
While not wishing to downplay the situation you personally are in, I can feel a little sympathy for your designer (though I accept he caused the problem in the first place). While he is answerable to the builder, and the builder is answerable to you, he is ultimately faced with the gamble of either paying to apply for planning permission and hoping it will be approved; or avoiding that gamble by paying to have the work altered.
If the worst came to the worst and your builder had to pay to alter it, are you sure they would be financially able? They could just walk away and the onus would then be on you to take action for recovery - you could end up throwing good money after bad.