They way that I, as a woman, got into YBU was to wait until I was a junior to transfer because so many freshwomen and sophomores dropped out for their Mrs degree. I can't remember where I picked this up but I did that and went there for my last two years of college. I loved the education, HATED the breed 'em young mindset. >:-(
The "no beard" rule didn't happen at BYU until the 60s. When I was born the President of the Church wore a beard. Every Church president for the previous 100 years had worn a beard. They can't claim that the "no beard rule" is a gospel principle.
However the "no beard" and "short hair" rules have proven a good way to control the student body. It will continue as long as it is useful.
Who attends BYUI and he has several large and very noticeable tattoos. I asked him how he got away with it and he said that no one has given him any issue about them. He was also a HUGE guy, huge. Maybe nobody wanted to piss him off. ;)
The anti-beard policy no doubt arose in reaction to the counter cultures embrace of beards. The church is run by businessmen, so it is natural that they want other members to look like businessmen.
The funny thing is how they enforce it. There is no "beard card" but you take the doctor's note to the student id office and get your picture taken with a beard. It has to be an obvious beard, too. That way, when teachers look at the roll, and when you show your ID in the testing center (the true enforcer of the honor code at BYU), they know it is legit.
This reminds me of attending YBU many years ago when you had to register for classes physically (not online) and they had a university nazi type that would stand by the students in line and tell them to go home if their hair was too long (hair over the ears for example), had facial hair, or for female students, if their skirts were too short.
That is allowed. When I was there (~10 years ago), there was a big commotion in the school paper because there were a few girls walking around with shaved heads. However, I don't think it violated the honor code.