You can use the word "but" as a transitional word, but it is recommended to not use contractions. Likewise, your sentence will make more sense by changing doesn't to does not.
The sentence we are completing is formed by two clauses with contrasting ideas. Thus, we should choose a transitional word that helps convey contrast. The best ones would be "however" and "but". We also need to pay attention to punctuation. If we were to use "however", we would need to change the punctuation. Therefore, to maintain the sentence the way it is, the only possible option is "but".
She tries to be patient but doesn't always succeed.
