I recommend you stay with Sony, Panasonic or Canon.
Your price range allows you a good selection. I would suggest that you also include a tripod, two additional high-capacity batteries and a sturdy case (like a Pelican 1500) in your budget. I strongly suggest you stay with either miniDV based or hard drive based camcorders. Do not consider DVD based camcorders.
Basketball games can last upwards of 2 hours. Burning the video from a whole game to CD will result in either very small window image playable on your computer or if you want to fill the computer screen or TV, the video image will be rather poor. CDs hold only 700 megabytes. You will want to burn to DVDs (most likely double layer DVD which hold 8.5 gigabytes).
I would also suggest that you reconsider your "no editing" position. Presuming your computer can handle it (we don't know what you have, so we don't know its capabilities), it is very easy to add credits at the beginning of the game - Who is playing, when the game was played, where the game was played... and at the end of the game, if you are feeling adventurous, adding player and coach names, your name and whatever else you want to put in that makes sense would be nice.
And you can cut out all the timeouts or whatever else you want to remove. Heck you *could* use whatever happens during pre-game or halftime to run credits over... This actually sounds like a lot of fun.
Warning: this is a fast game - keeping up will be a challenge. I learned a lot when I first video-taped a football game from the field. You get a really good appreciation for the skills the 20+ camera people have following the ball on TV...