Hello, I'm Keith, a Chartered Secretary amongst other things, and happy to help you with your question.
A Director of an UK Ltd company need not reside in the UK, nor, indeed, need they be an UK Citizen. There must always be one Director in a private company. A Director of a Ltd company is an employee per se, there is no way around this and must receive emoluments under PAYE arrangements.
The tax code NT is a direction to an employer not to deduct Income Tax (IT) under PAYE, so that gets rid of that problem.
The amount of profit for Corporation Tax (CT) within the company would be reduced by any emoluments paid to the new director. Thus the CT liability would be reduced. There would be no IT to collect and pass across to HMRC quarterly in respect of the Sark director.
I do hope I have put your mind at rest on this matter and maybe also expanded your Company Law knowledge.