Actually you have much more information than this. By pressing F11 you see the in game profiler aka Profiler1 which will give you a sense of what can be slow. If you press it multiple times, you'll see different informations. If you press Shift+F11, you take a snapshot of all this data in the form of a txt file called profile.txt you can send to us. If you press Shift F11 multiple times, you can send us multiple measurements.
It gives you for instance the number of triangles for 3D models, but also the how many milliseconds the renderer takes per turn or the simulation.
The lag shouldn't be affecting the game so much. It's not really network lag, that's the problem e.g you don't have a bandwidth problem since we're sending very little data. What you have is more of a ping issue where every player is expecting to give you news every 200ms and some fail to meet that deadline, so they have up to three turns to catch up (600ms)