There is no evidence for a sunken city on Doggerland, nor is there ever likely to be as it was inundated well before large settlements became a thing in this part of the world. What we know about the people who lived there suggests they were hunter gatherers taking advantage of what was a rich and diverse ecosystem.

We've known about human settlement on Doggerland for a long time, mostly through stray finds brought up to the surface by trawlers and dredgers. Latley Vince Gaffney et. al. have done some amazing work using very large seismic and sonar data normally used for fossil fuel exploration. These geophysical data can't resolve the relatively ephemeral Mesolithic settlements, but they do enable archaeologists to reconstruct the landscape before it was inundated, and enabled them to target coring for paleo environmental studies and to find evidence of human activity.

The hypothetical city in your question would almost certainly be identified by geophysics. Cities are large, and would be readily detected as anomalies in both the seismic and sonar data. Archaeologists would then take cores from the seabed to investigate the nature of the anomalies. Any long term human settlement would show up on these cores, both directly as anthropogenic material and indirectly as evidence of environmental impact.