Hi fishpan,
From my understanding, as discussed in-depth on [this forum post] by Thomas Pronk, is that you cannot get the exact time the browser flips. Here, Thomas also discusses the implications for timing on RT, I highly recommend you read this post and their article (mental chronometry in the pocket).
I don’t know anything about movie presentations in Psychopy/JS (or at all tbh), however, my natural intuition is that if the movie is fully loaded by participants prior to the movie’s display, there should not be dropped frames. Are your video files quite big?
A random side-thought, have you explored whether the frame dropping is occurring on specific browsers? As not all browsers are equal, it maybe advantangeous to restrict and only allow participants to join on your preferred browser, if you have not done so already.