5 round trips should be tops, even according to Swaty instructions, after the touch up you should strop and this should alleviate the harshness as well as refine the edge.
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I would recommend a Swaty, Apart, No. 00 Frictionite combination, Dubl Duck combination, Itsapeech, or one of the finer Carborundum Co. ones. There are almost certainly other barbers' hones in their class. I would agree that the Swaty is smoother than 12k.
According to the instructions that came with a NOS one that I picked up the side with the company name is coarser then the plain side. They also say to use as many strokes as necessary to get the razor to the level of sharpness that you desire. The non marked side being the finishing side. Before I had something finer in a stone I used a Swaty following my 8k Norton and it was good. The stones are better IME. YMMV.
You know for years I've wondered about the 4-5 laps deal with Swatys and have pretty much come to the conclusion that it's a combination of a marketing ploy (use our stones and it only takes 4 strokes!) and an attempt by Swaty to keep the user from going crazy and overhoning. I'm also puzzled by people who think nothing of doing 100 laps or more on a Norton 8K or a Coticule being afraid of more than 5 laps on a barber hone.
Since I'm firmly entrenched in the "less is more" camp (and the razors I touch up are already fairly sharp) that's why my routine is 5 laps and test followed by 3 laps and test...and as many 3 and test as it takes. If it ain't sharp it ain't sharp.