Kit or home remedy, you will not have swirl marks unless you are using a motorized tool and either applying too much pressure or leaving it in one spot too long. If what you are using is compatible with water, you might try using more water too.
Although it takes more time, it is easier to get a good result doing it by hand. Unless the lens originally looked CRUSTY yellow, and definitely not at this point, you do not need to start as coarse as 800 grit or lower. Those are only to remove the crusty yellow layer and even then, on a power tool may be too abrasive.
It's definitely not too soft to use 1200 grit in Canada at this time of year. Modest low temperatures should leave it hard enough as long as the removal rate does not create much heat itself. Just go slower so you build up less heat and more evenly refinish. Spend less time with the more abrasive compounds or discs and more with the finer ones. Move around a lot.
The home remedy is start out with rubbing compound or 1000 grit (or lower if crusty yellow as above) wet/dry sandpaper, then switch to higher grit wet paper, then switch to plastic polish or brasso/etc metal polish, then either toothpaste or a "polish" wax or sealant, but not a spray on paint/clearcoat type sealant which will just create a mess and need completely stripped off later. Whatever you use, the final coat should be a hand applied wax or sealant. Touch up with more polish wax/sealant or toothpaste plus that at each vehicle detailing session, or once a year if you can avoid parking with the lenses in direct sunlight.
The home remedy costs as much or more if you don't have most of the supplies because you end up buying more than needed to do a couple vehicles. It costs far less in the long run because you have more than enough to do a couple vehicles and supplies to do other things too... sandpaper, wax, etc, is versatile stuff, useful for far more than headlights.