A quick google search led me to this article: http://forums.ni.com/t5/LabVIEW-Idea-Exchange/Change-one-and-two-button-dialog-windows-do-they-do-not-block/idi-p/983941
You're right, the 1-button and 2-button dialog box uses the root loop, but the 3-button does not, so I could work around the issue by replacing my dialogs with the 3-button version. It works great now.
Thank you for pointing me to this "root loop", I had never heard of it before, and wouldn't have found that article that easily without this keyword!
eberaud's post in LV Dialog box holds Open VI Reference was marked as the answer