There may be no need... I stumbled across a fix that is working so far:
My VI front panel structure was:
Main Tab Control ("Tab Control 1")
Page 1: Tab control ("Tab Control 2") for sub process 1
Page 2: Tab control ("Tab Control 3") for sub process 2
Page 3: Tab control ("Tab Control 4") for sub process 3
Working with the blinking version in LV2009, I moved the tab controls for the sub processes out of the main tab control. I have no idea why, but the main tab control itself could be seen flickering every few seconds even though it was empty (and the tab controls containing the sub processes never flickered when they were outside the main tab control).
So I just created a new tab control (e.g., created "Tab Control 5") to serve as the main tab control and put the sub process tab controls inside it. Lo and behold... no flickering. I deleted the old main tab control ("Tab Control 1", now empty) with no ill effect.
My only hypothesis for this behavior is that somehow the tab control created in LV 2010 for Mac was either corrupted upon backsave to LV 2009 (or when opened in LV 2011 or LV2009 in WinXP). Perhaps it's platform specific; I combined the sub processes while working on a Mac, and the problem did seem to reproducibly platform specific (taking the buggy VI back to LV 2010 for Mac fixed it).
Good luck to those with similar problems! I'm not interested in starting a new thread at this time since the problem has gone away. I'll work in LV 2009 in WinXP exclusively (sigh). I hope there's enough keywords in this thread and post that people can find it via search engine.