Along with understanding of Bayes' theorem, there are quite a few additional topics which I feel should be part of a curriculum towards a "license to apply statistics with authority".
Even as an engineer, I can't overstate how important it is to learn Experiment Design, the way it is taught in a good biology course. You have to consider control variables, causation vs. correlation arguments, etc.
Another aspect of statistics, which is often not taught properly in engineering courses, is Fisher Information and the maximum-likelihood approach. Determining a probability function (and quantifying confidence in that function) from experimental data is vastly more complicated than generating data from a predefined probability function. If you want to challenge yourself: https://www.amazon.com/Detection-Estimation-Modulation-Theory-Part/dp/0470542969/
And then there is the art of selecting data to misrepresent reality, as covered by books that describe the methods used by the tobacco lobby to prove that "smoking is healthy".