OK, you're going to have to talk to your staff. You need a place to break. They're putting your safety at risk. Heat sickness is a real threat which can lead to death (I got it ONCE at the start of my career in '99, never again). They can at least use paper to block off a window or two on game days. You may be able to find a spot somewhere in the ballpark. My first baseball gig I was in the electrical closet! Sounds likes your park is small, but check if concessions has a walk in cooler. That's a great break place.
A lycra hoodie would be the quickest way to hide your neck (and keep your hair up), but I'm not finding an easy source. Wig caps work well for me, but usually tear. There are plenty of sports skull caps available, just look on amazon.
Sweat...I have one guy who swears by long sleeve under armor (or any moister wicking material). I tried it once and felt it made all the sweat drip off me and pool into the hands/feet. So I just go with plain old t-shirts. I put on a dry one every break.
Fans in the head don't work. Ice packs melt and add another heat trapping layer. A Kold Kollar might help a little, possibly while you break rather than while in costume.
Hydrate yes. I've found electrolyte pills (with potassium) help me a little during the really hot days (the potassium helps with post game cramping).
But yeah, you need a little corner to yourself where you can change shirts during breaks, take at least the top of the costume off, and be near a fan. Make friends with the stadium ops people and see what you can find. Good luck!
I've used this in my own attempts before- its cross stitch mesh and is easy to come by.
The least expensive semi decent bunny is the rubies one. That's actually a super cheap price (normally $300-$400), they'll be gone by easter I'm sure. The hands are just mittens and the feet aren't that great, it's molded/poured latex/foam and has no helmet inside (arms straps) but if $ is your main concern that's your best bet.