Is it a good idea for a nonnative English speaker to have native speakers help make the English in statement of purpose perfect?
It's definitely a good idea (in fact anyone in your situation should do so), I have seen plenty of people around me in my graduate school who had done so. There even exist professional services specialized for this task (it's a pretty big business in some countries).

Regarding your concern that the jury might think that you have been "cheating", forget about it:


by providing a flawless statement of purpose despite being non-native, you are showing the jury a lot of motivation;
if the statement of purpose was used to assess your true level of English, they wouldn't ask for TOEFL/IELTS;
when writing research articles later on, you will still often have a native speaker around to answer questions.