That's a really interesting topic. Although I've bought each book, magazine and newspaper I was able to find within the last 15 years, I couldn't pick up much information about Maximilian's illegatimate children. By the way, he didn't seem to have any illegitimate sons, because in books there has unexceptionally been the talk of "daughters".
It seems to be confirmed that Max used to have dinner with these daughters. Also Brigitte Hamann mentioned that in her biography about Elisabeth.
Strange enough, Duke Max turned to be some kind of maverick in his later years. He lived in his rooms in the Maxpalais in Munich and didn't want to see neither his family nor most of the servants. Marie Redwitz, Duchess Amélie's lady-in-waiting, saw him for the first time when he was lying on his deathbed.
Marie Larisch - well, it's up to you if you believe in her stories - told that Marie of Naples wanted to see her father shortly before his death. When she was on her way to him, she met "one of the duke's illegitimate daughters he loved so much" on the stairs and talked to her. If this is the truth, Max's illegitimate daughters were allowed to enter the palais and they got to know the duke's legitimate children. Besides, it seems like the duke favoured his illegitimate daughters also in old age and has kept close contact to them over the years.
I've never come across any names or birth dates. That's a pity!