In my opinion, B is just paying for expenses of an employee which are regularly reimbursed by company A. Hence it is an expenditure in the books of A in US and not in the books of B in India and hence I believe that company B need not deduct tax of an employee out of payments made. However as employee is in India working for company A, it may constitute PE of company A in India. Tax is to be deducted by company A and shall be deposited in India. Company A shall obtain PAN and TAN in India and comply with TDS provisions.
Further I suggest that as the said employee is director in company B as well, proper documentation shall be maintained through out the year to prove that the said employee is working for company A and not rendering any service to company B in India as it will be quite important for Transfer Pricing in India as authorities may ask Company B to charge its parent company A for services rendered by an employee for company B.
I have gone much beyond your query. Sorry about that but such issues are interconnected and to be treated very carefully throughout the year with robust supporting documentation.