Convergence issue may appear for various reasons. For example, it is not just the length of the central region, but the length of the electrode extensions is to be enlarged to check if that improves the convergence behavior.
In relation to the electron difference density and electrostatic potential, the behavior of these quantities and the SCF convergence speed might require enlarging the screening region, which is a part of the central region between the electrode extensions. That is especially true for non-metallic systems such as intrinsic or low-doped semiconductor, because screening is not efficient in this kind of systems.
So, the convergence and electron density behavior are dependent on your system of study. In any way, you should first find the right system parameters and computational settings to converge the system before looking at the electron density and other quantities. You may still compute them as a weird behavior of these quantities in the vicinity of electrode extensions may suggest where the actual problem is.