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I have a similar problem with horse show photography, being inside the show ring with the competitors, I find a monopod too clumsy and slow, as I have to move around quickly, and stay safe. So, in dark arenas I use on camera flash as a fill, with high iso on my 5DM3. I occasionally add a second off camera flash if I can find a place where it won't interfere with the horses or get blocked.
I highly recommend the 600 EX RT flash, as it will reach to 200mm, and easily expands to an off camera system with either a future second flash, or transmitter. I found having that extra reach, which the 580 EX II lacks, makes a difference. Check the Buy & Sell for a used one. Its worth the investment.
Would it be possible to have an assistant hold the flash for you during the game? I have seen brackets which allow for the flash to sit to the side of the camera, however it might not make much of a difference being only a few inches away from the lens. So, for the sake of this limited experiment, on camera flash might be ok to try, and will give you the flexibility you need.
I use Powerex rechargeable batteries in my flashes, they are powerful, and by using high iso the flash output is reduced, which gives you a very fast recycling time, without an external battery pack. Enough to get 2-6 shots in a sequence.
You won't have to change your camera settings, just add the flash, and set it to high speed sync. The flash should give you a better overall exposure, but you will still have to do NR with a 1DM4 at very high iso.