VictoriaCalledTori is right on this; the Greek version of the name begins with the letter Chi, the letter with which Jesus' title of "Christ" ('Christos' in Latinised Greek) also begins. And "Christ" isn't pronounced "SH-rist", is it? The same logic applies to the name "Chara", QED.
I agree with seraphine_eternal. Since it's strictly a Greek name, the only "correct pronunciation" is the Greek one - and that's Kara (or, Khara). PS: I'm not sure what it has to do with Chara but anonymous, let me tell you – as someone who speaks 5 languages, all Indo-European – that English is the simplest of them all by far. What you describe has nothing to do with complexity of English language and everything to do with English speakers's bad habit of applying their native pronunciation to foreign words without thinking. (no offence intended).
Let me start by saying this is a goofy name, but the meaning is pleasant. English is one of the hardest languages to learn and if you don't know the "Ch" is pronounced three different ways: "K", "Sh","Ch". However, The Greek letter "X", known as "Chi", equals plain "C", therefore Chara=Cara. You decide.
The name Chara has the same first five (four in Greek) letters as the word "character" (which itself is from Greek), so the initial "ch" should be pronounced in the same way. Note that the "k" sound in English is only an approximation; it's more like a soft "k" or a guttural "h". For English speakers "k" will do, but never "ch" as in "Charlie" or "sh" as in "Shirley".
Note also that, in ancient Greek, the accent is on the second syllable, but that will probably sound rather unnatural for modern English. The "a" in the first syllable can be pronounced as in "father" (my preference) or, in a more anglicized manner, as in "hat".