DD, I would just consider this the normal type of stray spike and is exactly the type thing we are guarding against by being watchful enough to get both down to zero in 12 hours and to do the qualifying week. Since you are way out at the end of the 3rd phase of fishless cycling though, I'd definately do a 90% (down to the gravel) gravel-clean-water-change and refill with conditioned, temp-matched tap water and recharge ammonia back up to 5ppm (and recharge baking soda if you are doing that.) It may take a couple days to get its numbers settled back down to something understandable after a big water change like that but hopefully you'd then find yourself ready for the qualifying week or alternatively that yes, you have some problem with your nitrite-processors and they need a little more build-up time.
Meanwhile, examine your stocking plan with an eye to whether any of the species will not be there (for instance, if neon tetras need to wait the obligatory 6 months from tank start or you want to wait for better water for the more expensive centerpiece fish or something like that) and if that's the case and you will be stocking only at 50 or 75% of the inch per US gallon guideline then you may be able to consider just skipping over the final "sticking problem" (where nitrite (or, rarely, ammonia) keeps showing a trace at 12 hours, but then goes away shortly thereafter. Sometimes in those cases we tell people to just go ahead and do the big water change and keep a watchful eye for spikes.
Does all that make sense?
~~waterdrop~~