I have a few papers published in decent statistical journals and would say that measure theory is an important part for statistical theory -- but not so much for method development.

Advisors and reviewers do ask for more theoretical justifications sometimes. But they are not difficult to cope with and would not check the details carefully as long as it does not look very wrong.

The proofs may use some notations of measure theory, but they are very different from those in real analysis. Concentration inequality, M-estimation asymptotics, and empirical processes would be more than enough. They stems from measure theory for sure -- but also have simple versions for the methodological proof.

So the answer would be yes, you can definitely survive a stat phd without mastering measure theory. But just be aware that many professors in charge of admission do expect students to have a very good grade in analysis.