The reference to KLM/Chase suggests to me that this is a Flying Blue award ticket. Is that the case? If so, I would have thought that cancelling the first leg while leaving the rest of the itinerary intact should in principle be possible. If it is a paid ticket rather than an award ticket, they might be more reluctant to do so but there is no harm in asking if this is what would work best for you.
Bear in mind, though, that if you book the Precision Air flight separately, you will have no protection if your miss your KLM flight in Nairobi due to delay or cancellation of the Precision Air flight. In addition, you may well have to collect your luggage and re-check in Nairobi (and get a transit visa to go landside for that purpose). In your shoes, I would personally be a bit wary of doing so, as booking a last minute flight to SFO from NBO could be very costly. If the only solution is to book a separate ZNZ-NBO ticket, I would personally consider an overnight in Nairobi to be on the safe side
While the prospect of spending many hours in Nairobi by getting an earlier KQ flight would not fill me with delight, I would give some consideration to asking to be rebooked on an earlier KQ flight so as to keep it all on the same ticket.
Also,if it is a Flying Blue award, KQ may well tell you to speak to Flying Blue rather than themselves but see how it goes.