It's difficult to provide a meaningful analysis based on the facts set forth in your post. To enforce your rights, you have probably figured out that you will need to consult with an attorney one-on-one. There are several law firms in your area that can assist you with this matter.
Reckless endangerment is in the Tennessee Code at section 39-13-103. The law says that reckless endangerment can be a misdemeanor or a felony, so for punishment purposes, the facts matter how the reckless endangerment happened. However, if you are convicted of it, it does not show up as one "type" of reckless endangerment or the other. Plain and simple, what happened to cause the conviction only matters in sentencing not with how it shows up on paperwork.
You could get a copy of the paperwork from the court to show that it was driving related, but then they would know you were charged with DUI.
Depending on the facts of when you received the conviction, whether you have any other convictions or served any jail time, might make it possible to expunge or remove this from your record, so you may want to talk with an attorney to see if that is possible.
This answer is for informational purposes only. It is not intended as legal advice. Nothing in this answer is intended to create an attorney-client relationship. I encourage all people to seek a consultation with an experienced attorney on any legal matter.
They are not contacting the Dept of Safety when they are doing a background check.They are requesting the check from the Tennessee Bureau of Investigation. If you get a record check on your drivers license more than likely it will show Dui reduced to Reckless Endangerment. The same thing will probably be the case with a county check of the incident. It will probably be hard to get the TBI to change it but i would send them a copy of your driving record or a copy of a certified judgment from the court where you pled. Good Luck!