To start, here is the section of Mein Kampf where Baumann is mentioned:

Thereupon, I decided to remain, but the discussion seemed to proceed without anything of vital importance being mentioned, when suddenly a ‘professor’ commenced to speak.

He began by throwing doubt on the accuracy of what Feder had said, and then, after Feder had replied very effectively, the professor suddenly took up his position on what he called ‘the basis of facts,' but before this he recommended the young party most urgently to introduce the secession of Bavaria from Prussia as one of the leading points in its programme.

In the most self-assured way, this man kept on insisting that German-Austria would join Bavaria and that the peace would then function much better. He made other similarly extravagant statements.

At this juncture I felt bound to ask for permission to speak and to tell the learned gentleman what I thought. The result was that the honourable gentleman who had last spoken slipped out of the room, like a whipped cur, without uttering a sound.

The mysterious 'professor' was Dr. Adalbert Baumann, a professor and secondary-school teacher, born in Karlstadt in 1870. In 1918 he founded a party called the Demokratisch-sozialistischen Bürgerpartei München (Democratic-Socialist Citizens' Party, Munich). In early 1919 he stood in the Landtag elections but was not successful.

It is worth noting there was a huge number of tiny fringe parties in post-war Germany. Bavarian secession was a reality, though it would soon fail, but Baumann advocated joining the nascent Austria rather than standing alone - he seemed to blame Prussia for the disaster of the war.

As far as can be inferred, he then joined the Unabhängige Sozialdemokratische Partei Deutschlands (Independent Social Democratic Party of Germany; USPD) which had been founded as an offshoot of the Social Democratic Party. There, he made quite a name for himself - Richard Sheppard claims that there are multiple reports of 'violent, class-based verbal clashes between rank-and-file party members and one Professor Adalbert Baumann.' Baumann advocated uniting the 'leaderless' Munich middle-class to defend against 'Bolshevik rule.'

Baumann left party politics in 1920. After Hitler seized power, Baumann wrote to Goebbels - confirming his debate with Hitler:

Fate, too, has led me through the hopeless foundation of the Civic Party into that small circle where a debate with Herr Hitler on Austria's anschluss drew your circle's attention to this gentleman.

The purpose of Baumann's letter was to advocate for his concept of a simplified German language, called 'Wede,' which could be used in German colonies and as an international lingua franca. His idea was never taken up, and Baumann fell on the wrong side of Bavarian Gauleiter Hans Schemm, losing his job in the process. He was ejected from the NSDAP in 1937.

The date of his death is uncertain. It is listed on his German-language Wikipedia page as 1941, but it is unsourced - I would therefore be cautious with accepting this, as he has been confused in the past with Adolf Baumann (such as in Othmar Plöckinger: Unter Soldaten und Agitatoren). Plöckinger asserts that the entire debate was made up, but unless Baumann was writing on a level of imperceptive sarcasm in his letter, this seems unlikely.

As to the actual content of the debate, it seems hard to ascertain anything more than the above generalities without access to the Bundesarchiv - but even then I'm not sure how much we could gain.

If anyone here is proficient in German, I would like to welcome them to expand on my answer, as I am not confident enough to fully refer to the 1933 letter. Furthermore, if anyone has access to Hitler, Mein Kampf - Eine kritische Edition, it would be helpful, as I believe this may have a little more on the debate - it is, however, prohibitively expensive and not available yet in my university library.

Sources:

Hitler: 1889-1936, Hubris by Ian Kershaw (1998).

Mein Kampf [My own copy is the so-called 'Stalag Edition,' which I am not confident provides a completely accurate translation, but the gist is there].

Vom Trommler zum Führer: Der Wandel von Hitlers Selbstverständnis zwischen 1919 und 1924 und die Entwicklung der NSDAP by Albrecht Tyrell (1975) - especially page 195. Available in German here.

"Artists, Intellectuals and the USPD: 1917-1922," by Richard Sheppard in Literaturwissenschaftliches Jahrbuch (1991). [The article is in English.]

Baumann, Adalbert: Ein Sprachamt in Europa by Gerd Simon (1996). Available here in German [the site may upset your browser, but it is safe].

Baumann, Dr. Adalbert in the Bavarian Parliament Archives. Available here.