imagemeister Loc: mid east Florida

Sony RX 100 II - has f1.8 lens and you can use 1.1-2X Clear Image zoom to get closer with out losing light.

BRdancer Loc: Carlsbad,CA

Thanks for your really helpful suggestions. I'll follow up on them. :)

BRdancer Loc: Carlsbad,CA

Thanks for your helpful suggestion. I'll check it out. :)

BRdancer Loc: Carlsbad,CA

Thanks everybody!!!!! You have been so helpful. I have lots to work with from your guidance and will take all into account. Now I have work to do to check out these cameras. Thanks again!!!!

sirlensalot Loc: Arizona

IMO, you will need a DSLR or a MILC to capture the action, adjust WB, and shoot in RAW mode to minimize PP.
Suggest avoiding kit lenses for your purpose and go with an inexpensive prime like a 50/1.8 or ideally an 85/1.8 which gets you as close as the 4X zoom of the 1300b but tilts your budget in a not-so-friendly manner.

The OP does not mention if flash is used. I am assuming no flash permitted?

For my money, I would stretch the budget and look at a Sony a6000 ($448) with the Sony 50/1.8 O.S.S. (stabilized) lens ($248). I use the a6000 - 50/1.8 combo for indoor sports under some pretty sketchy lighting and it performs extremely well. I am able to get what I need shooting JPEG's.
Saw the whole kit at Buy.Dig with 16-50 kit lens, 32GB card, table top tripod, and camera bag for $598, but beware of grey market stuff. Most reliable for me is B&H. They may price match. I doubt the kit lens will be sufficient for your application, but just guessing.

ddonlewis provided a great option for your budget as well.

If small and easy to use is the main criteria, suggest Sony RX100 that fits your budget, or the even faster, but larger Canon G16 which I believe has a viewfinder and shoots 9FPS. In your price range, it's all about minimizing trade-offs to fit your needs as close as possible and of course the budget.

Maybe call B&H and ask for their advice as well so they can fit your budget to what they offer.
Good hunting