Was the reckless endangerment originally filed as a DUI? If so, then it would have been coded as a driving offense went the court sent notification to Olympia (DOL). A Reckless endangerment can involve a motor vehicle (which gets reported to DOL) or not involve a motor vehicle (which does not get reported).
The only way I know of to get it changed with DOL is to convince the prosecutor who filed the charge to re-file the charge as a non-driving reckless endangerment. However, if you have already pled guilty and been sentenced you would have to move the court to vacate that conviction before the prosecutor could refile it as non-driving.
My guess is that you will need a lawyer to do this for you, and even then, the prosecutor may not agree to it--particularly if it was a DUI originally. Thus, you may spend more on legal fees than you would save on insurance.
Now, if you were NOT charged with reckless endangerment involving a vehicle, but it was coded that way as a mistake by the court clerk, you might be able to move the court to amend the abstract of conviction that was sent to DOL due to a clerical mistake.