This seems to be getting more complex, not less. I am guessing that the employee is effectively asking the employer to extend a business trip as a BIK.
Seems to me that you may be conflating deductibility with BIK. Again, if the overseas visit is necessary to secure ongoing/more business with a particular client, then it seems prima facie to pass the W+E test. I suppose that there is a risk that a statutory disallowance might arise under client entertaining provisions, but it could still be a deductible journey so long as entertaining was not the sole/main purpose. E.g., if the directors have a five-hour business meeting but then take the client for a meal, seems to me the T&S relating to the business meeting should still be allowable as the fundamental purpose of the trip (But does the client need to see both directors, or really only one?)
If the company is bearing the cost of providing the directors with holidays on top as a BIK, then that is allowable to the company if it can be argued that it is part of the director's/directors' remuneration package, and incentivises them to do work to increase business, etc. (Disallowing it does NOT prevent its being a BIK.)
Assuming this is not Salary Sacrifice/Optional Remuneration Arrangements, then I should normally look at the MARGINAL (additional) COST as the starting point for BIK. While the extra nights in the hotel could amount to 7/8ths of total hotel bill, I doubt that the extra cost of the flight would amount to 7/8ths. Doesn't make sense. Again, this is for the BIK, NOT for the CT relief.
If the employer is contracting with the airline / hotel, then this is the company's liability, and I do not see why the employees might have primary NI exposure. I would expect the company to contract if it were motivated as a business expense. Ordinarily, one might worry about Employees' NICs only if the employer were settling the employee's own personal pecuniary liability - the employee had contracted personally, but the company then settled the employee's personal bill.
I trust this goes some way towards addressing your queries.