You mention that the drinking was due to "life stress". I wouldn't mention that AT ALL to anyone!! Did you already say that to your supervisor? They will be all over that, as obviously it's an example of you being unable to handle stress properly and using alcohol as a crutch. They'll try to get you into some type of treatment.
You say "what's the big deal" since you don't have a problem. The big deal is that you are looking at an assessment for several hundred dollars (I don't see how your supervisor can claim it will be free), then the assessor will decide you turn to substances (alcohol) to deal with stress, so you need some type of treatment. This will cost you, or your insurance if you have coverage, and you won't be allowed to work so you'll be out an income. That can go on for months before you are cleared to work. Also you'll have to start attending AA meetings, and submit to random drug tests, which will end up costing thousands of dollars. You will be labeled an addict or alcoholic, since you are enrolled in the monitoring program. No one will care why you're in the program, just that you are. You will have a TERRIBLE time finding a new job due to being in the program. You'll have stipulations which makes it even harder to find a job (no administering or even counting narcotics, no overtime, no floating, no night shift). You'll do this for 3 years or 5 years, every single day you'll wake up with your heart pounding because you're afraid that you've forgotten to call in to see if you've been selected to test, and you're afraid now it's too late (even though you have until 2pm every day to check in, every single day you'll have a moment of fear: did I check in today??). It's hell.
If you can, say that night of drinking was spending time with your dad.. You didn't realize how late it was, he kept ordering you a new drink, you didn't realize how strong they were making the drinks etc. Just don't admit to drinking due to stress.