In The Elder Scrolls V: Skyrim, the Imperials no longer have strictly Latin sounding names. Now they have some individuals with Italian sounding names, such as the Avenicci family. The name "Avenicci" (along with some others, presumably) are where this problem stems from. The Dragon Alphabet lacks an equivalent to "c", apparently because it could be replaced "k" or "s", but now there's no character for the "č" sound (made with "ch" in English and "ci" or "ce" in Italian).

In short, I want to know how Dragons would write words from outside there language? Would it be like Italian where letters are added for the new word (jeans, as Italian doesn't have "j" in most dialects) or like Japanese where they make do with what sounds they have (furuto for flute, since there is no "l" in Japanese)? And in the case of the latter, what would be the closest equivalent to "č"? Using "s" instead of "c"?

Good question!  It would probably be spelled as literally as possible in cases where the sounds exist.  "Cyrodiil" might be spelled like "Syrodiil", etc.

I think we will have to come up with a letter combination that stands for "ch".  "ts" or "tsh", perhaps?  Then "Avenicci" would be spelled "Avenitsi" or "Avenitshi".  There are currently no Dragon words that contain "ts" or "tsh" so adding this won't affect the pronunciation of existing words.

Yeah, saying it aloud, "Avenitshi" sounds almost the exact same. Should there be a new rule that "tsh" be used for "ch" in the case of loan words?

Sure, I think that's the best way to go about it without messing with the existing alphabet.