Firstly, Finno-Ugric languages have nothing to do with Baltic languages (at least at present in Latvia), not even with Indo-European I believe. Although there is a Latvian dialect called Latgalian or "Livonian dialect" it shouldn't be confused with the Livonian language.
Firstly, saying that two languages have not been proven to spring from a common source is a far cry from saying that they "have nothing to do" with each other. We know that prolonged language contact leads to significant changes in the languages involved, up to and including a complete rearrangement of their syntax. (See the article http://en.wikipedia.org/wiki/Language_contact
for examples.) It's unthinkable that Baltic and Finnic languages haven't influenced each other; the only question is what exactly these influences are.
Second, it's an open question as to whether Finno-Ugric and Indo-European are ultimately related or not. Actually, the ultimate relationship of all
natural languages is an open question, since there's of course no way to prove a negative. But hypotheses linking these two groups have far more support than most other high-order proposals. (See http://en.wikipedia.org/wiki/Nostratic
for more details.)
Secondly: I read about the German monks in a museum in Sigulda in Latvia once. But I may be mistaken. I see that Wikipedia and a book about the Baltic countries of mine doesn't describe it that way.
I can't imagine what good docents at Sigulda were thinking when they wrote that unless it was the fact that German monks were to reduce Latvian to writing
. But creating a written record of a language isn't the same thing as "inventing" it any more than photographing a baby is equivalent to giving birth to it.