I agree with the advice that you've already been given. We had a very similar thread and discussion a little while ago about tattoos. The reality is that even if the firms themselves have no particular issue with facial piercings, there may well be concerns that their clients might, and no firm is going to risk losing or in some way offending clients over something like that. I do know plenty of lawyers whose look is in some way unconventional. One of my best male friends is a barrister in London with long hair, which you don't see very often at all in male barristers, and I know various solicitors and barristers with a range of tattoos, piercings and hair colours. But these are all pretty established lawyers working in areas where their clients care very little about these aspects of their appearance. The top law firms that you're talking about are more likely to either hold adverse views on facial piercings and similar, or have clients that they fear might hold adverse views. Ultimately it is important for clients to have trust in their legal representatives, and different things to go into that, including the lawyer broadly looking like the client expects a lawyer to look. The clothes that I wear do not in any way impact on my ability to run a trial and act in the best interests of my client, but if I turned up to a conference in a vest and shorts I wouldn't blame my client for questioning whether or not I'm the right barrister for them. Image does matter for lawyers.
Is any of this fair? Probably not. I don't think your desire to get a lip piercing tells us anything at all about your potential as a lawyer. But it is the reality of the situation and this is a competitive industry where you should be looking to gain advantages, no matter how small, over your competition, and should equally be looking to avoid disadvantages. That doesn't mean conforming to stereotypes in every respect, but in my view it would mean not getting the lip piercing at this stage. As has been said, I just don't think it's worth the risk of it having some sort of negative impact on your applications.