I'd suggest creating the dictionary properly and not trying to "fake" it with strings or literals.
e.g.
Code:
var doc = new PdfDocument();
var page = doc.AddPage();
var privateDict = new PdfDictionary();
privateDict.Elements.Add("/DocID", new PdfString("122996605"));
var defaultDict = new PdfDictionary();
defaultDict.Elements.Add("/LastModified", new PdfString("D:20231212073902-05'00'"));
defaultDict.Elements.Add("/Private", privateDict);
var infoDict = new PdfDictionary();
infoDict.Elements.Add("/Default", defaultDict);
page.Elements.Add("/PieceInfo", infoDict);
doc.Save("out.pdf");