How to overcome transistor heat in slayer exciter?

Have you tried finding the current through the coil? Find the current through the coil. You can either simulate it with a spice package or place a meter between the coil and transistor. Once you know the current use the equations located in this online textbook in chapter 11 Inductance and Magnetic energy (Yeah it requires math).

Then find the magnetic field through the coil (you'll need to know the area of the coil and the turns.) Then calculate what kind of energy you need at the input coil. I'll give you a hint as to what you'll find. Your coil is generating a magnetic field at a certain frequency, keep in mind the magnetic field is going to 'attenuate' by the the inverse cube of the distance (magnetic field=1/(distance^3)). Meaning if you had 1 unit of magnetic field at 1cm, at 2cm you will have 1/8th of the field, at 3cm you will have 1/27th and so on. By the time you get to 10cm you will have 1/1000 of your original field so you will need ~1000x the current or you will need to redesign your coil.

This is also why there will forever be companies that promise wireless power but you never see any products beyond the charging pads. The lightbulb demonstrations use phenomenal amounts of power even with a directed antenna(coil). The best one I've seen is this one Wireless power transmission. You can changed the antenna/coil however and it can help. Look up Helmholtz coil if you think that might work for your application. If you want you can parallel several BJT's if you want them to get less hot.

Ideally you would use something more efficient (and necessarily more complex) than a rudimentary slayer exciter, but to easily improve the current circuit you need just a proper power transistor that you mount to a heatsink.
The 2n2222a comes in a TO-92 plastic package, can dissipate a pathetic 625 mW of heat and is only rated for a current of 800 mA.

You want something much beefier, like a 2n3055, which can dissipate 60 W and is rated for a collector current of 15 A. It is probably also the most easily available power transistor.

A 2N2222A is totally inappropriate in this application. That's a small signal transistor. You need a power transistor. There are many to chose from, but something like a TIP41 is something you can probably find easily.