Why not use NPS?
NPS asks about recommending, once, and aggregates the answer. Client happiness asks how happy the client is with you now, per client, per month, so you can act on the individual answer. It is a steering instrument, not a marketing statistic.
Who should ask the question?
Someone who is not the founder. If you ask, clients round up. If the account manager asks, the answer is honest and the meter belongs to the team, which is where it should live.
What if a client refuses to give a number?
Then you have your answer. A client who will not give you a 1 to 10 is not a 9. Treat it as a 6 and have the conversation.