That rules out the K701/702 then.
If you can get yourself some Q701s and get yourself some Annie or 712 pads from AKG spare parts then you will have a phone that can do it all.
If not the Q701 on it's own is around neutral on the bass so if the material has bass it won't accentuate it but play close to what was recorded.
The K612 pro which is a lot cheaper is a better balanced headphone but lacks the sense of space and atmosphere the Q's present. In return you get a warmer, fuller and bassier sound which has a good stage but not great.
With headphones there's trade offs. Warmer fully phones tend to lack the open airy soundstage of of a headphone that is thinner and less warm.
The only way to get in the middle is go for something like the 65th Annie or Q701 with Annie pads that add that sense of space but still compared to the Q701 stock they still lack the sense of air.
DT770 is an excellent phone, it lacks air but has great bass, warm with amazing accuracy for positional audio and has a very large soundstage. The trade off is openness due to them being completely sealed.
A900x does what the 770 do with less body so you can hear details a little easier..less soundstage width but more air due to the thinner notes.
These all meet your requirements it just depends what you can afford to loose out on.
Hope this helps.