Firstly, if you want the best results, start shooting in RAW. It gives you more latitude to play with the images after the event so that you can retrieve more information in the darks & lights. With Jpeg, what you see is pretty well what the camera gives you and obviously you want more. If you are unsure using RAW, shoot both jpeg & RAW and you will have a safety net.
Secondly, I know that someone said earlier that shooting at 1/1000-sec is over kill, but if your images are un-sharp then you obviously need to shoot faster. You say that you are shooting at 1250, is that 1/250th second, or 1/1250 second? if it's 1/250th, then go up to 1/500th at least. 1/1000 isn't going to hurt. Are you shooting hand held or tripod? maybe a monopod might help, just to add a little more stability.
Does the KP have IS? it's not a model I've ever used, I'm a K-1 & K-3 III shooter. If it does, use it, but not whilst on a tri or monopod, it's for hand held use & it will help with camera shake, meaning that you can shoot a little slower hand held. If it doesn't have IS, maybe it's time to upgrade to a second hand K3 or K3 II, there should be a few on the market with the new model now out. Image stabilization would be pretty essential for shooting sport & with 2 daughters playing soccer it sounds like you are going to be doing it for a while. Have you tried shooting on a high burst rate? and panning with the action?
An extra stop of ISO will help in giving you an extra stop in either shutter speed or aperture, it's up to you to decide where you want to use it, speed or DOF, but if you are shooting wide open at f2.8, f3.5 is really not going to add a lot. Maybe look at f5.6 and push the ISO up further.
Lighting seems ok, no issue with white balance. Remember grass is around middle grey as far as light values are concerned, so shooting in the middle of the light meter should give you what you need. BUT again, shooting in RAW is going to give you a whole lot more latitude to pull an image back. Hope this is of help.... good luck.
PS, just because I'm classified as a junior member by the forum, doesn't mean I don't know what I'm doing. I taught photography for over 30 years & am a master photographer..