iMessage shows ICE instead of contact
iPhones have a neat feature that allows you to put your emergency contact information on the lock screen so no one has to unlock your phone to access it. You can set it up through the Health App on your iPhone.

Since your Contacts won't be accessible on your iPhone without your passcode, this is a reasonable alternative. 

If you set this up, you could theoretically delete the entry in your Contacts that is listed as ICE, thereby fixing your problem with the contact name showing up as ICE.