Is putting a silly name on a diploma a bad idea?
Yes, with conditions.
If it is a military law school then any aka might be viewed as having a lack of self confidence in your given name, and thus a potential lack of confidence elsewhere, not a good thing for a military lawyer. But, maybe "Devil Dog" might work for a JAG position.
If you are planning to be a divorce lawyer then maybe "Get It All" might draw in more clients.
Generally, yes it is not a good idea. But that depends upon where you intend to be in the law field.
Are diplomas submitted in law/graduate school applications, or just records containing GPA and the like?
In my experience, both with conditions. The more arrogant and disdainful for the masses of common folk that a department chair is, the more that they might request up front or before the interview a full disclosure of transcripts of classes and grades (sent to them directly from the attended colleges). Some might be casual but still request these records to be sent to them from previous colleges that you attended. Some might just request that the attended colleges send them a single sheet copy of your graduation diploma. Hope for the easiest, prepare for the not so easy. It does not stop there. I once applied for a master's level degree program and they required that I attend two (2) weeks of pre-approval classes wherein the students were required to attain and maintain a 4.0 average for that class before being admitted into the program. I once applied for a doctorate program which required the applicants to take an hours-long computer software based psychological evaluation. Be prepared. Know ahead of time what you will tolerate and have other degree/school/university options already researched and considered.
"I personally feel it is a harmless bit of self expression"
There are consequences to all actions.
If you are seen as silly or a clown once, by one person, then the average dimwit (which seems to be average in itself) person, might in their desire to play along, long after the joke, verbally assault you to get more laughs, thus transferring their actions to other mentally challenged persons viewing the event. This can snow-ball quickly. This can last for the rest of a person's life. It should be stopped with all necessary (and in case some dimwit needs it to be said, "legal") force. This seems to be common among humans and even among some animals. Avoid being perceived as silly.