anata doesn't have to be overtly personal, it can also be overtly impersonal, but nnnnnnnnnnnno, there's nothing inherrently wrong with anata, just its ambiguity is something to be avoided.
i'd still use her name, but i defer to the actual translator/interpretor

So I could change anata and put the katakana for her name in the exact same place as anata would go in the sentence?

あなた（Anata） is a formal you, When used between couple, thats another case. あんた（Anta)works as well

If you want to be impolite, yes, it will work.

Around here, it`s downright insulting.

and i was just about to give up

Give up? On what?

あんた is pretty common in 関西 region. many girls use it in particular. I wouldn't consider it rude at all between people you are already on friendly terms with. i wouldn't use it in polite situations of course.

still would avoid using あなた even among friends.
anyway, olsson88- did some asking around today and came up with this.
you have two translations
translation 1) 最も大切なのはあんたの幸せ
motto taisetsunanowa anata no shiawase
translation 2) 一番大切なことはあなたの幸せ
ichiban taisetsuna koto wa anatano shiawase

translation two, i changed the mottomo to ichiban as the vast consensus was that it sounded more natural. mind you, both translations are correct grammatically and mean the exact same thing. now, when asked to produce the same translation for "the most important thing is your happiness", all the exchange students produced translation 2 with little to no differentiation. When asked which they would prefer in a letter, all Japanese picked translation two as being easier to understand- HOWEVER, when asked what they would write to express the same feeling, the Japanese came up with something like this:

わたしはあなたが幸せになることが一番嬉しい
watashi wa anata ga shiawase ni naru koto ga ichiban ureshii

so anyway,
there you have it. my endorsement goes to the japanese's 文.

That`s the thing - it`s regional. It has a different connotation in other parts of Japan. The only examples I and my family can come up with are condescending usages, even between friends. That may be local, but it`s use as just another version of anata is fairly specific to the Kansai area.

But that only works well in a relationship context. The original poster didn`t specify the context, it`s just an assumption. For all we know, it could be a letter to a sibling about their choice in schools, etc.
You certainly wouldn`t use that sentence in a letter to a colleague who is debating whether to change jobs.

by the way, i picked your sentence for comparison because it was the only other sentence that was correct.

and as another 関西代表, my advice is to use people's names whenever possible. and even when you dont know a person's name it really doesnt matter because more often then not you don't use anything at all. i just really cannot emphasize this enough to beginning learners of japanese: dont call people anata

also, i'd caution against thinking stuff is ok just because you've heard it.