I don't think it would be appropriate to use a sentence fragment such as also known as or which should be on their own, because they don't refer to anything in particular. It would be correct to say it like this: James, also known as Jim, was at the meeting (or something like that). But if you simply write James, also known as Jim, I would be asking myself, "What did James, also known as Jim, say or do?"
As someone who writes fiction, however, I find it annoying that grammar checker often "corrects" things I put inside quotes. People do use fragments (rightly or wrongly) when they speak to one another, and so portraying realistic dialogue will require you to do this. I have also noticed that it dings me for certain contractions, even when I have the correct contractions feature turned off.
And don't even get me started about spell checker's complete inability to recognize scientific, medical or technical terms or words that are not common but are still part of the English language.