Measure your case for height available to install a good air cooler.
160mm gives you very good options.
Less than 145mm and you might as well keep the stock cooler.
8400 runs stock at 2.8 with a 4.0 turbo. Turbo is generally for one thread when other threads are not heavily used. cost is $180
8600 runs at 3.1 with 4.3 turbo. $215.
8600K at stock runs at 3.6 with 4.3 turbo. $245.
You decide.
But, if you use a z370 based motherboard and overclock a 8600K conservatively, you will run at 4.7 on all threads.
Overclocking is a misnomer for the new processors. It consists of upping in the bios the all core multiplier from the stock 36 to a higher number, letting the voltage remain on auto.
Hard to forego a potential 30% benefit.
And, you...