When trying to create a minimal reproducible example as suggested by trincot and ggorlen, I took another look at my original code and realized that the cause of the issue was having a helper function invoke the Puppeteer instance methods without binding this when calling them. Whether or not I was using Puppeteer within a module turned out to be irrelevant.

E.g.,

// I was doing this
function callMethod (func, ...args) {
   func(...args);
}
callMethod(browser.newPage);

// And what I really needed to do was something like this
function callMethod (obj, method, ...args) {
   obj[method].call(obj, ...args);
}
callMethod(browser, 'newPage');

During my original troubleshooting, I replaced the first couple calls to callMethod() with a direct call to the respective Puppeteer instance method but left all of the other calls to callMethod() intact, obscuring the fact that callMethod() was the true culprit all along.

Lesson learned: Always make a minimum reproducible example, no matter how "simple" you believe your existing code is.