Just the fact that you were even asking that question however raises another one. What does it actually mean for a translation to be "influenced" by a doctrinal tradition? I agree it happens in various ways and to various degrees. But there's an interesting aspect where I also think this is a good thing.
Surely we don't want to make the text say something it doesn't, but we do want it to say whatever it does say with words we can understand. Our theological frameworks do, over time, shape our vocabularies and even use of language in general. Words and even grammar structures come to have associations they wouldn't have otherwise.