Based on what you said, you should not have any entry on your CII or NCIC rap sheet. My guess is that whoever did the background check found this through a search of the court records which are public record. However, your lawyer should be able to answer this for you.
Basically, a rap sheet entry occurs when you have been booked. Since you were not arrested, this should not have happened. However, if for some reason it did, then the dismissal did not get reported.
Talk to your lawyer about whether or not you are entitled to have a certificate of detention not arrest if the case was never filed or whether you can get a finding of factual innocence if the case was filed.
Having said all that, if the check was run by a private employer, it may be a violation of the Labor Code to take action on an arrest that did not result in a conviction. If the arrest is not open, then they cannot require that you disclose it and cannot punish you for failing to disclose it. If it is a law enforcement check or a federal matter of some sort, then you may have to disclose and explain.
In any event, talk to you lawyer and good luck!
You were not "convicted" of the charge if you never pled guilty to the charge or went to trial on the charge and were convicted by a jury. You may have been technically "arrested" without knowing it if you were detained by police and given a citation. This may constitute and show as an arrest on your record. The charges were "pending" against you from the time they were filed with the court until they were dismissed.
It sounds like you were never notified of the charges or had knowledge they were pending until the background check was done. Since a year had gone by and you had no knowledge of the charge, the case would be dismissed by the Judge, because of the lack of prosecution by the DA. In other words, they filed a misdemeanor against you, but never brought you to Court within a year period of time. Congrats on the dismissal!