I don't believe the Deathrunner can use the crown of conquest. This is down to the wording of crown of conquest;
"Friendly CHAOS units do not have to take battleshock tests while they are within 6inches of the bearer".
The crown says you "do not have to" implying there is a choice therefore an argument can be made that you are using the crown of command and its not a passive that is always in effect.
An example of this is - A deathrunner with Crown of command is next to a 40 man clan rat unit that gets shot by weapon X and 10 clan rats go down. Now battleshock phase happens and you have a choice, either take a battleshock test or use the crown of command which triggers the deathrunner to use the artefact.
I still think that this needs some further clarification