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.