Yes you are completely justified in wanting to file a complaint.
Five months is a VERY long time to wait to notify you of the fact that the DMV was in possession of your GC ..not to mention the additional issues it caused of having to cancel a trip and apply for a replacement card.
So again, yes you are completely justified in wanting to file a complaint.
Unfortunately from a legal perspective, there is not really any civil or criminal action you can take against the DMV or its employees because there was not any law broken.
That being said, you can still file an internal complaint with the DMV.
A supervisor would then take a look into the situation, see who the employee/employees were that were involved in initially finding the GC and notifying you, and then hopefully take some sort of measures to reprimand the employees involved.
I hope that helps! Please let me know if you have any other questions. Also, please take a moment to rate me using the stars at the top of the page, as it's the only way experts are compensated for our time on the site. I do not receive a salary here at JustAnswer, and am only compensated for my time after receiving a rating of 3 stars or more.The question won't close after rating, so you can still ask follow-up questions after. Thanks, ***** ***** it!