I don't like Klasse, I think it looks cold and artificial and as you saw in your experience it magnifies defects in the paint. So you really have to remove all the defects before Klassing and that can be very frustrating and time consuming. Plus when you remove defects you remove paint. I prefer to remove what I can easily (which is most) and hide the rest, you just can't do that with a synthetic product like Klasse or Zaino. Topping it with a carnuaba helps the look but wont hide the imperfections.
Klasse AIO is a good oxidation remover, but other than that I have little use for Klasse.
I wax every 2-3 months. When I wax I polish again with something mild and with nice oils like P21S GEPC and then seal with something compliant like Meguiars NXT, Poorboys EX or EX-P. Then maybe depending on my mood top with a wax like #16 or P21S. This is very safe as there is nothing really here to remove paint.
Now removing swirls DOES remove paint, and I only do this once or twice a year. By hand not much is going to happen though. If you perfect your washing routine swirls will not be as much of a problem.
Honestly? You need a Porter Cable DA Polisher or make an appointment with a detailer to have them compound out the swirls and you can go from there. Its just too hard to remove swirls by hand.
Clay might work, also try a vinegar rinse to see if that will help.
Compounding with a buffer will help as it will round off the edges but there isn't much that can be done.