Is there a way to set the default vertical exaggeration in the profile design view of a line?
Yes, the CIVIL_SET_DEFAULT_EXAGGERATION variable must be defined.
This variable can be defined on the user level (per one machine), by going to File>Settings>Configuration>Configuration Variables and setting the value here
or at the workspace level, where all users opening a dgn in a particular workspace will see this change, by setting the value in a cfg file. Below is a snippet taken from the "_Civil Default Standards - Imperial.cfg" found in the default directory (C:\ProgramData\Bentley\OpenRoads Designer CE\Configuration\Organization-Civil) of the Bentley workspace.