The ideal lens in this situation is probably a 70-200 f/2.8L USM70-200 f/2.8L and the (non-IS)older Mark I USM (non-IS) version, andmight fall inside the Mk 1price range on the used market might actually fall within the price range. TheThe lack of IS is less of an issue with sports, where your shutter speed is liable to be up and over 1/focal_length to freeze the action, anyway.
There are also three fast prime options you can consider, all of which are known for their sharpness and speedspeedy autofocus performance.
The EF 85mm f/1.8 USM (nicknamed "Mr. Basketball" on the dpreview Canon lens forum because so many people want a cheap basketball lens)
The 85/1.8's identical cousin, the EF 100mm f/2 USM
The EF 135mm f/2L USM. As an L, it has better contrast and build quality, and 135 is the center of the 70-200 range.
However, the longer a prime, the more restrictive its framing can be on usage. I have a 135L, I shoot events with it all the time, but it can be frustrating not to have zooming capability, especially if your own mobility is restricted.