Do I have to both capitalize and italicize the term "bildungsroman" in a sentence?
No. You don't need to either italicize or capitalize this word.
This word is a common noun, so there is no need of capitalizing it.
Some people may think that this word should be capitalized because it is a German noun and in German all nouns are written with capital letter, but this is only rule in German grammar, so it is not required to capitalize it when writing a sentence in other language than German.
I am not sure about italicizing this word. Some people use italics to write words or longer fragments when they are in other language than the whole text, but there is no such grammar rule.