I want to translate:
"I gave my little sister a doll for her birthday".
I do not want to use 私 but imply that I did it, so the right particles are crucial.
I've come this far, but I'm not sure about the particle は which I chose basically because it was the most likely one that was left.
誕生日{たんじょうび}は妹{いもうと}に人形{にんぎょう}をあげました.
My original choice was 誕生日に but then I would have two にs.
How can I say "for occasion"?
There is no problem having two に particles in one sentence. Your original choice is most correct.
Using [event]+に is the best way to express that something will happen for [event], and [person]+にあげる happens to also be the best way to express that you are giving something to [person].
誕生日｛たんじょうび｝に妹｛いもうと｝に人形｛にんぎょう｝をあげました。
As @Eric mentioned, に is the correct choice, and there is no harm in having two of them in this sentence. In addition to that, you could use には to emphasize that it was specifically for her birthday instead of some other occasion.
誕生日には妹に人形をあげました。 → For her birthday, I gave my little sister a doll.
Note that you can also use [贈]{おく}る for giving a present.
This isn't literal but it seems natural:
妹の誕生日に人形をあげました。
To say 「誕生日に」＞「誕生日は」 in OP's context is very Japanese-as-a-foreign-language-esque.
There is absolutely nothing wrong or unnatural in saying 「誕生日は」. In fact, 「は」 would be a very natural choice among native speakers.
To attach 「は」, the word does not have to be the grammatical subject of the sentence.
