Decided to ignore budget and just look at Corsair cases as a matter of curiosity.
Corsair 540 Air: CPU cooler clearance 170mm, Dimensions (mm) 332 x 415 x 458 (W x D x H) -
too wide
Corsair 500r: CPU cooler clearance 165mm, Dimensions (mm) 206 x 520 x 586mm (W x D x H) -
too high and not enough clearance
Obsidian 550d: CPU cooler clearance 180mm, Dimensions (mm) 220 x 530 x 495mm (W x D x H) -
just within height, should be good
Obsidian 650d: CPU cooler clearance 185mm, Dimensions (mm) 229 x 546 x 521 (W x D x H) -
too high
Obsidian 750d: Maximum CPU Cooler Height 170mm, Dimension (mm) 546 x 236 x 558 (W x D x H) -
clearance good, too tall
Then looked at Thermaltake:
Thermaltake Soprano Tower Case: CPU cooler height limitation: 175mm, Dimension (H x W x D): 460 x 220 x 510 mm -
seems good!
Thermaltake Urban S31: CPU cooler height limitation: 155mm, Dimensions: (H x W x D) 500x 206 x 520 mm -
no cooler clearance
Thermaltake Overseer RX-I: CPU cooler height limitation: 177mm, Dimension (H x W x D): 535 x 220 x 580 mm -
too tall, no clearance
Thermaltake Urban S71: CPU cooler height limitation: 160mm, Dimension: 534(H) x 213(W) x 584(D) mm -
too tall, no clearance
Thermaltake V3 BlacX: CPU cooler height limitation: 160mm, Dimension: D473.0 x W190.0 x H445.6 mm -
no chance
So 2 out of 10 so far, and not much with the required cooler clearance despite the size. I'll look at micro-ATX only now and see what I can uncover.
edit:
AeroCool Dead Silence: CPU cooler clearance 190mm, Dimensions (mm) 265 x 381 x 411 (W x D x H) -
winner!
AeroCool Qs-202: Height limit for CPU coolers 157mm, Dimensions 378(H) x 200(W) x 470(D) -
no clearance
BitFenix Colossus M: CPU cooler height at 160mm, Dimensions (WxHxD): 250 x 330 x 374mm -
not a chance
BitFenix Phenom: CPU cooler clearance 170mm, Dimensions (WxHxD): 250 x 330 x 374mm -
winner!
BitFenix Prodigy M: CPU cooler clearance 160mm, Dimensions (WxHxD): 250 x 404 x 359mm -
no clearance
In-Win BUC 101: 170mm (CPU HSF), Dimensions: 496 (H) x 200 (W) x 508mm (D) -
not sure which is depth or height, could be tight?
Lian Li PC-A04B: CPU cooler height:140mm, Dimensions 188mm x 386mm x 460mm -
no clearance
Lian Li PC-V355B: CPU cooler height: 100mm, Dimensions: (W) 283mm x (H) 263mm x (D) 386mm -
no clearance
Parvum S2.0: Maximum CPU Cooler Height: 152mm, Dimensions (WxHxD): 275 x 305 x 420mm -
unfortunately not 
Silverstone SST-SG10B Sugo: Limitation of CPU cooler: 165mm, Dimension: 354mm (D) x 220mm (W) x 295mm (H) -
no clearance
Xigmatek Aguila: CPU coolers up to 180 mm tall, Dimensions: 265 x 403 x 390 mm -
winner!
Xigmatek Gigas Cube: CPU cooler height: 160mm, Dimension 278(W) x 322(H) x 396(D) mm -
no clearance