So, how important is it to promote their real name?
You can promote your unique nickname or your real name, or both. It is your personal choice and no one will be able to choose one or the other of these two/three items.
Or you can safely work under a pseudonym?
You can run without my real name. Actually it is not some supernatural phenomenon because each person might have a hundred reasons to work under a pseudonym. It is only a name and everything. Name, though the Master of the mirror or Ghost Hunter is not how you gain skills. Of course a well-chosen pseudonym would be a good item, but however well it depends on your imagination and taste. IMHO.
As a really this are the customers? There are examples?
Adequately treat. If the alias is adequate and does not cause fear/laughter/rejection/other senses and/or emotions. Everything must be carefully, seriously and professionally.
PS I mean not the use of the nickname, and for example, more relevant/eye-catching/memorable name, and the name to leave now.
Then what is it? If the name remains real, and the name becomes the alias, then all General rules. Qualitatively do their job and all will be well, and all the rest is nonsense.