Truman - good luck with your work situation. It sounds like your goals and your work's goals for your appearance aren't that different: you both want a tidy professional appearance. It just seems that they don't get that it's possible to have that with anything but a smooth face or a very short beard.

I'm sure you have played lots of different situations around in your head, but if you went preemptively to your boss and brought up the fact that your main priority is to keep it tidy and then show her a picture of an exceptionally manicured 3-month beard and let her know that something like this is your goal, do you think she would react positively? Maybe not if you're getting grief for your current look, which I would classify as very tidy indeed.

Looks great by the way - I hope you get to keep it.

You're also getting a wide variety of advice on this and may not want any of it. So feel free to ignore mine. I sit in an office surrounded by other scientists and engineers dressed in clothes two or three steps up from pyjamas - so take my advice with a grain of salt anyway.