Most of the noise associated with 3D printers come from two factors: stepper(motor) activity & fan noise. By default, the S1 pro for instance, has stealthchop enabled, which is a feature that's become default for many printers. Back in the Ender 3 day and before, stealthchop wasn't an option on the older TMC drivers that came default. The difference is night & day.

As this is not a viable option for reducing motor noise, we can look at the sound generated by belt motion on idlers. Changing to a linear rail system may remove some of this noise, but that involves rebuilding your entire S1, and since the K1 is already running on a rod system, your noise won't be coming from there.

You can change out the fans, but stock fans perform at roughly the same noise level as what can be bought aftermarket.

Although completely out of the scope of your inquiry, but the most effective solution, is the process of noise isolation in the form of an enclosure. Your K1 will already isolate much of its noise, as it's enclosed. You could build a foam enclosure to prevent noise from escaping from all sides. Foam is inexpensive & easy to work with.

Consider how vibrations travel through materials. If you have a large-rigid surface that the feet make contact with, it may act as a diaphragm & translate motion into sound. Keeping your printer on the smallest surface possible can reduce this noise

K1 printer noises