With that kind of budget, you can get top-end air coolers. If you don't mind splurging now, you can get a beefy one now and plan to keep it through upgrades. Options that I can think of:
- Noctua: NH-D15S, NH-U12A
- Scythe: Fuma 2 (I suggest fan upgrade as well)
- ID-Cooling: SE-207-XT (basically Fuma 2 with louder fans)
- be quiet!: Dark Rock Pro 4 (163mm, YMMV)
- Thermalright: Le Grand Macho RT, Macho Rev B
- Corsair: A500 (162mm, YMMV)
YMMV: Deepcool Assassin III is among the best, but it measures at 165mm while your case states 164mm height CPU cooler. Ditto with Deepcool AS500 and NH-D15.
Not too experienced on AIOs, but AFAIK the difference between units in the same size is usually down to how good the stock fans are. I'm sure someone more experienced can chime in on this. That said, high-end air coolers are usually better than 240mm AIOs.
More budget options: Mugen 5, Dark Rock 4, DR Pro 3, Cryorig H5 Ultimate. I would say that a common 120mm single tower cooler can do just fine especially if you do no to small overclocking, like Hyper 212 Evo/Turbo, Gammaxx 400, Cryorig H7.