First, no one can push security to do anything, no matter how high up they are. The company can't afford to be seen as anything less than absolutely flawless in that department.
Second, even though 5 years seems like a long time to you, it does not seem that long to them, particularly since the most recent isn't 2 years old yet. As far as they are concerned, you are still showing a pattern of breaking the law and you are doing it in a way that historically has shown to be a virtually unbreakable habit.
Third, the unfortunate fact is that you are the person that you appear to be on paper to anyone but you and your mother.
It is entirely possible that you could win an appeal, but it is more likely you won't. There just isn't enough reward for the risk they would be taking. If you were a 20 year nuclear veteran that had proven your worth to the industry and had a unique set of skills that were hard to replace, you would have a fighting chance. So far you have nothing to show that would make anyone think you were that valuable.
There is a lot of competition out there for good jobs, making it a buyers market for the employers. They can and will select the candidates that pose the least risk and offer the biggest reward. If you go to the Farmers market to buy some apples and finds stacks and stacks of them, are you going to look closely for bruised ones and pick the best looking or are you going to just pick without checking?
If you have your heart absolutely set on working in Nukes, by all means keep pushing. But first you have to follow the advice you have been given. Go to rehab even if you don't think you need it (nobody that needs it thinks they do at first.) Be squeaky clean.
Proving you are a good person is much harder than proving you are not. Good Luck.