So I'm going to be starting ballet classes in a few weeks, and the dress code is a leotard and tights, and I'll obviously have to tuck in this outfit. So my question is is there a method of tucking that is safe for exercising and sports without pain or damage.

I would say to try taking a tube sock and a underwear/pantyhose elastic and cutting the sock to make a tube and put the elastic through to make a gaff. It might help but other than that I can only say to take some painkillers and try to hold out.

Have you thought about sports tape?
I did it every day for 3 years, worked well, never had a real issue if you prep and look after the area afterwards

I push the bits into my abdomen, fold the excess skin around the shaft and tape to avoid skin being pinched, and then use a sock tube and pantyhose elastic to keep it in place between my legs. I have trained martial arts tucked like this for a year now, and have not had a single problem so far. With the tight clothing you will be using, the chance of things getting untucked is even lower.

I use silk tape, by the way. I have found it to sit well but also come off well with no irritation.

Kinesiology tape. With sock tube and elastic band it will be problem because the lines would be very visable with wearing leotard, and it really looks at least strange if not ugly... With kinesiology tape and a little practice you can make things really flat, bikini level flat.