Originally Posted by
Frozen_Feet
That's the canonical name and best description of what they do. What would you suggest? "Order of Eibon" is flavorful, but doesn't sound like something a military Division would officially call itself.
To give a rough real world analogy: medics are made to swear the Hippocratic Oath, but instead of being "Order of Hippocrates", any real world medical company is just a medical company. I have no other gripes with your write-up, again, it's just a naming issue. Gotei is a military force, and military forces tend to have blunt informative names for their departments. Even the canon follows this, with each Division being called either by a number or their function. ("Kido Corps" doesn't leave much room for guessing, for example.)