If that is the case, then I would recommend the OP look into Masters programs where they can take two semesters of NON-measure theoretic real analysis in their first year (along with the usual two semesters of Casella & Berger statistics and applied regression/design of experiments). If they have *no* experience with mathematical proofs, they should certainly not be taking measure theory.
However, looking at the undergrad major for Statistics at UW, it looks like they actually do require a semester of Real Analysis as part of the major? https://stat.uw.edu/academics/undergraduate/major
However, the OP is a "data science and statistics" major so that might be different than the BS in Statistics.