Verdict
The K1 wins on raw price ($200 less). The P1S wins on failure rate (3.0% vs 6.9% on stock K1 firmware), AMS ecosystem, and out-of-box reliability. Klipper-tuned K1 (input shaper recalibration, custom slicer profiles) closes the gap to ~4.5% — but that's a weekend of work.
Buy the K1 if: you enjoy tuning, want Klipper customization out of the box, $200 saved matters more than $200/year in waste reduction.
Buy the P1S if: production work, multi-color via AMS, want fewest hours per week spent debugging printer issues.