There is absolutely no reason why you could not do it all by yourself.
You have first to call BA and get the booking split into 2 PNRs. You cannot do this step online unfortunately. No one is going to ask you for the credit card details - only potentially if you are the credit card holder. (I won't advise what your answer should be at that point).
Once you have your own PNR you can UUA online. The payment of extra taxes and surcharges can be done on a new credit card, it does not have to be the same that was used on the base booking.
Am sure your CEO has better things to do than meddle with your company travel - and you may not want him to know you upgrade your company bookings.