That gear is exactly what you need.
Re "wrong bike...."
Really depends on your plans and expectations. Tires are definitely the limiting factor here simply because of availability. After that, it's suspension and brakes. As your pace increases those start to matter a lot.
This bike will do fine in red group and even into the lower end of yellow group, but if you're going to stick with this (as noted above, you'll want to), then something a bit more current will make your life easier. Something like an RC390, CBR300, Ninja 300/400. All are similar to the 250 but with a bit more power (nothing crazy) and better parts availability. Moving up from there, the SV650 is the go-to recreational track bike... spares forever, lots of development, nice blend of power and torque. Built SVs compete with my Aprilia in MotoAmerica.
If I were in your shoes I'd ride it for now, learn some things and use that knowledge to decide what the right bike for you really is. I'm assuming that since you intentionally bought a 250 to take to the track you're not the stereotypical "A 600 super sport is a beginner bike" guy who lusts after horsepower. Little bikes are way cool and will teach you a ton.
Re tires....
Are the tires hard, like hockey pucks, especially at the edges? That's what happens when they age. They vulcanize. You won't crash on them but you'll find the limits of grip sooner rather than later. If they're still reasonably soft, they're good to go.
If you're set on replacing them, I found some tires for you and the links below show them in stock. The Bridgestone Battlax BT46 is, according to the manufacturer, the successor to the BT45 and that's a known-good tire (do some searching here on Ninjette). Or just go with the Shinkos. They'll be okay.
If you're up for taking your carbs out and dunking 'em in an ultrasonic cleaner, I assume you're mechanically adept and can change tires yourself.
https://www.amazon.com/Bridgestone-P...302240&sr=8-18
https://www.rockymountainatvmc.com/t...orcycle-tire-p