This is getting to the point of absurdity...
WHAT I AM DESCRIBING IS NOT FLASH PHOTOGRAPHY.
The "flashes" are not direct; they are hardly noticeable. And any player who is at all focused on the game won't notice at all. Most fans don't notice until I point it out to them–the flash is too indirect and its duration is too short.
What I'm describing is the poor man's version of what the professional major league photographers use. Instead of mounting true arena lights in catwalks and rafters, I suggest putting strobes 12+ feet up on lightstands in the corners of the arena, with the lights aimed at the ceiling and wall (AWAY from the players, to further reduce any "direct" light).
These flashes are triggered via RF, not an optical sensor. There is no flash emanating from the camera itself. -> no direct flash. no distraction.
That said, if you've got the money for a nice ƒ2.8 lens go for it. The lens is more versatile than the lights, and even good entry-level lights won't be able to make up for slow glass on a body that doesn't have great high-ISO tolerance.
Ideally, you'd be shooting ƒ/2.8 (or better) on a D3 with true arena lighting... but seeing how money is a limited resource...