I did have the viewer configured so that the layer being identified was the top-most layer. The problem only occurs when I try to use a mix of feature and dynamic operational layers -- if all the operational layers are dynamic, the popups function normally.
I've looked into the Identify Widget in the past, but, as best I could determine from the documentation/forums, it does not work with secured services. So, unfortunately that is not an option.
For now, I'm just going to make all the operational layers dynamic (viewer is for next week's primary) and test the 3.0 prerelease separately.