I cannot tell you much about Beethoven, but I can tell you a little bit about German.

I don't see the original German of the letter but I assume the word being translated as "science" is Wissenschaft. Wissenschaft in today's German has a much broader meaning than "science" does in English. In English, science tends to be the exact sciences and the life sciences, possibly including the social sciences, as opposed to the more subjective humanities. As a non-native speaker, I hesitate to say exactly what it is, but my impression is that Wissenschaft often means more theoretical knowledge as opposed to practical knowledge, but it's a sort of knowledge production broader than merely the natural sciences.

Duden (the most reputatable German dictionary) defines it as:

1. (ein begründetes, geordnetes, für gesichert erachtetes) Wissen hervorbringende forschende Tätigkeit in einem bestimmten Bereich.

(a well-founded, ordered, secured) knowledge-producing research activity in a specific area

2. jemandes Wissen in einer bestimmten Angelegenheit o. Ä.

someone's knowledge of a specific matter, etc.

Compare that with Merriam-Webster's "essential definitions" of science:

1. knowledge about or study of the natural world based on facts learned through experiments and observation

2. a particular area of scientific study (such as biology, physics, or chemistry) : a particular branch of science

3. a subject that is formally studied in a college, university, etc.

(I should note they include broader definitions in the full definitions, but I think this covers everyday usage). Notice that the German definitions are about knowledge production generally, while the American definitions include the method of knowledge production (experiments and observation).

I was made aware of this because I saw a professor's tote bag for some conference which I think was called "Geschichtswissenschaft" (History-Science) and was quite confused about what the "Science of History" was all about. The field I studied in as an undergraduate is called "the History of Religions" in America, but the same field is known in German as "Religionswissenschaft" (which is also how "Religious Studies" is translated). It's about knowledge production but is not necessarily tied to the scientific method. You have "Bildwissenschaft" (Picture Science) which is a very broad reading of Art History. Nietzsche's Gay Science (in German, Die fröhliche Wissenschaft) is about poetry and literature and this was a common saying not just in German but in other languages during this period.

Perhaps a translation like "study" or "research" rather than "science" might make clearer the broad meaning that Beethoven likely had in mind.