I assume you have the Lumagen set to output a fixed resolution to the projector. In other words, the Lumagen is having problems syncing to the Oppo's output.
If you haven't already, you should verify that both the Oppo and the Lumagen have the most recent firmware. If they do, then contact Lumagen about the problem.
Edited to add: one workaround to try is to set the Oppo to always output a fixed scanrate (e.g. 1080p60). Since the scanrate wouldn't change between the menu and the feature movie, there should be no dropouts.
Of course, this would be only a temporary workaround until you get the sync problem resolved, since it uses the Oppo's scaler instead of the Lumagen's,