I'm surprised to see so many people here recommending a 50mm prime lens. "Zooming with my feet" may be fun to say, but in a situation where you're in a crowded, enclosed space and you're trying to stay on the fringes and out of people's way, it's just not realistic. You'll wear out your welcome very quickly.
And while it may be tempting to use apertures wider than f/2.8, I think that you'll run into focus problems with your fast-moving subjects slipping out of the razor-thin focal plane.
The recommendations of a 70-200mm f/2.8 are also suspect, especially since you said you wanted to stay around $300. A 70-200 may be great for tournaments in a large venue (assuming you don't mind having a log-sized lens attached to your camera), but in the average dojo you're often going to wish you could back up about 15 feet through the wall.
I believe the person who recommended the Tamron 28-75mm f/2.8 was spot-on. You'll have a slightly-wide angle at 28mm allowing you to easily capture a pair of combatants squaring off, but with a range of up to 75mm you'll still be able to get a tight shot without being right in your subject's face.
You'll need some luck to find one for $300, but if you can stretch your budget to $350-$400, you should have no trouble finding a nice second-hand copy. And if you don't already have an f/2.8 constant-aperture lens, you'll wonder how you ever lived without it.
And in most situations, f/2.8 should be fast enough, as long as you don't mind raising the ISO. I know that on my K-x, ISO 800 and 1600 are good enough for large prints. While there is some noise present at those levels, I do not find it objectionable and it kind of resembles a fine film grain. And if you convert your images to black and white, the grain is actually very attractive.
I've never shot with your camera, the K200D, but I used to have a K100D and I wouldn't hesitate to use it at ISO 800 or 1600.
Last summer I had a chance to shoot my nephew and his father at Tae Kwon Do with my K-x. I took my Tamron 28-75mm f/2.8 and my Pentax DA 55-300mm. I mostly shot with the Tamron at f2.8, 1/320 sec, and ISO 800. For the Pentax 55-300mm I used ISO 1600. If I were doing it over again, I would probably just take the Tamron and shoot at f/2.8, 1/400 sec and ISO 1600.
There were windows along one wall, but it was a gray, overcast day, so there wasn't sunlight pouring in. And I would almost rather not have any windows since mixed lighting sources can make it difficult to get colors and skin tones perfect.
Shutter speed of 1/320 was generally fast enough to freeze the action, but there will sometimes still be some motion blur at the extremities during moves. This is not necessarily a bad thing since it imparts a sense of the motion.
Here are some of the shots from that day. All of the shots are linked to the full-size 12MP image (minus a little cropping), so pixel-peeping is available. No noise-reduction was applied to the full-size images. Click on the image to download the full image file.
BTW, the grain on these shots is actually a little more than I typically see from my K-x. But it turned out that my shutter speed was too high so most of the shots were dark, which meant that I had to boost the exposure quite a bit in Photoshop. Since I shoot RAW, this wasn't a big problem, but it did result in slightly more grain than is typical. That's why if I were doing it again, I would just go ahead and shoot at ISO 1600 at f/2.8.