I don't think you'll find a 200 or 300 lens for ~$50, at least not a decent one. If you're going to be close, I think your best bet is to get a 135mm f/2.5 or f/2.8 manual lens. You can find those on ebay for around $50 and they take decent shots. I've had 3 different ones and they've all taken good shots. They're all full-frame lenses, so even if they're accused of being soft wide open or some such, they still work well since the smaller sensor in our cameras uses the "sweet spot" of the lens.
If the light is bright, you could get the Pentax DA50-200 for ~$200 online, plus there's a rebate. You can also get a Tamron 28-300 zoom for ~$250 (or less, if you can find the non-DI version - I paid $140 for mine but I think that deal is long gone).
Here's a shot I took at an NBA game from the other end of the court using the Pentax SMC-A 135mm f/2.8 lens.
I was probably 20 rows up at the other end of the court (~150 feet) when I took this shot. This should give you a feel for what kind of reach this lens will give you. Check out the full sized shot at my Flickr site (see the Houston Rockets set). For your $50 budget, I think this lens (or any of the 135mm primes) is your best bet.
Good luck!
<edit> Re-reading that your shoot should be dim and overcast really makes me recommend the 135mm since you'll never get f/2.5 or f/2.8 in a 200mm or 300mm lens for under $1000. Just get one far enough in advance to learn how to meter it, and then just focus on a spot in the track/road and let the cars come to you. Lots of sports were shot with manual focus lenses.