Find a retail store where you can handle all three. See which one feels best in your hands and to your eye. Use the through the lens viewfinder, don't judge them by using the lcd screen as a viewfinder. I'm assuming you have no system investment already. I would fa er the D7200 but I've been shooting Nikon since 1969.
Forget the kit zooms unless you're outside. Get a 35mm f1.8 and an 85mm f1.8, both should run about
$ 700.00. Use the pair indoors with shutter priority auto and auto ISO as suggested. Shoot a lot, electrons are cheap.👀
I shoot a D600 and have been using a pair of Sony 64GB cards with no problems. I have just started using a Lexar 128GB in place of one 64GB card with no problems to date. I shoot stills, RAW + jpeg, as well as video.
Sounds like you have a wonderful trip to look forward to, take as much backup as you can and have a ball!
How much light is on the playing floor?
How many foot-candles of light and how even is the lighting across the playing floor?
What kind of lights illuminate to area and what is their colour temperature?
Do you have a speedlight or similar flash? If you do, just shoot with the flash.
Just use the lens on a Canon body.
For your half frame i suggest the 35mm f2 as your prime, the 70-300 is a good addition and won't break your bank or back.
Look for Nikon FM2 bodies and AI lenses. The FM2's were my 35mm workhorses. I could never afford F2 or F3 at the time.
A great little lens, I shoot it on my D90 as a normal and on my D600 in FX mode. I do not mind the corner fall off, its what I grew up with in the film age. Get the lens and go shooting.
Forget the zooms, get a 20mm f2.8 Canon. Cheaper and sharper without extreme wa distortion.
Junk it and get an HP or Canon. After my experience with an Epson C66, I vowed never to buy another Epson printer.
Think of the old 300mm f2 Nikkor on a tripod using the lens tripod mount, add a Nikon F2 body with motor drive and 250 exposure back. Thousands of frames and the rig was solid as a rock. And now Nikon has widened the front body flange to make the mount even stronger.
Metal lens mounts to metal framed bodies are not usually a problem. If you want a rig that supports the body as well as the lens, check with Kirk Enterprises or a professional. Cinema supply house
Use the viewfinder and turn off the rear display. Then shoot a burst. That will tell you if your card is slowing you down.