I have completed the Headhunters' trail which follows the same route as far as the Pinnacles. Both treks involve staying overnight at Camp 5. Camp 5 has limited capacity. If the Park HQ is telling you that there are no slots available it is quite likely that it is because Camp 5 is full. If that is the case, then you may have difficulty finding a tour operator that can take you. However, tour operators often tentatively book accommodation and then release it a month before the date if they don't require the accommodation. I don't know whether that is what happens at Camp 5, but if so, you may find that there are slots available one moth out from your date. However, while to my be able to hike to Camp 5, I believe that you still need to go with a guide to climb the pinnacles. Have you thought about that?
In fact, I'm not sure what the limiting factor is as far as tour operators go. I don't think that anything puts a practical limit on completing the Pinnacles climb. It is more likely that cost is the limiting factor. The cost of taking one person would certainly be high.
I don't think that hanging out near the ticket booth at the park entrance has any prospect of offering you a solution. Completing the Pinnacles climb isn't just "a walk in the park". It involves significant logistics. It is unlikely that you would be able to put everything in place in time even if you happened to find someone at the park entrance interested in doing the trek.