The Mazdaspeed kit you have actually consists of Eibach springs and KYB shocks; very similar to a KYB AGX/Eibach spring combination. This kit is not “notoriously” harsh.
Setting 1 on the shocks is about the same as a standard shock absorber.
You will probably still find it “harsh”, and thats really due to the springs. If you want lowering, you will generally have to accept a stiffer spring; Eibachs are around 30% stiiffer than stock (as is the case of virtually any -30mm lowering spring). The Mazdaspeed Eibachs differ a little from the other Eibach springs in having a slightly stiffer rear spring. Replacing the springs with a standard spring, accepting a jacked up appearance, and you will probably have a softer ride.
Another factor; what people sometimes think as “harsh” is actually the rear bumpstops bottoming out on back roads, giving a jittery ride. Many an owner has spent large amounts of money replacing shocks, but keeping lowering springs, only to find there is no change. This is just a feature if you retain standard length shocks, standard top mounts and standard bumpstops.
My guess is, unless your shocks are leaking all over the place, fitting the Koni or KYB kit will see no real improvement.
If you want a soft ride, revert to OE spec shocks, springs and new bumpstops. If you want a decent ride AND go cart handling, then the kits from Performance5 are hard to beat; very stiff springs, with carefully matched shocks achieve an unexpectedly smooth ride. But its going to cost you.
If you can spanner yourself, cheapest option for you is to change the springs for standard spec; stick with original Mazda, as pattern springs either tend to sag, or are specced too long. The “Mazdapeed” shocks aren’t really any different to the AGXs (except in some small details), so can take a standard spring. renew the bumpstops while you are at it.
If you are paying someone to take your suspension on and off, now is not the time to penny pinch, given that the labour on fitting shocks and springs is the same as fitting just springs (or just shocks); ie. about 4-5 hours.