I've solved this, so I'll describe the solution in case anyone else wants to do this.
In my GUI, I had to enable rotation manually by using rotate3d. On a mouse drag, this function writes the camera azimuth and elevation to the current uimode's ModeStateData.textBoxText property, which is then displayed at the bottom left of the current figure. ModeStateData is a struct with a field textState set equal to 1. Setting this field equal to 0 turns off the az/el readout.
To do this in a gui, you'll have to get access to the uimode object handled by rotate3d. From a blog post on undocumentedmatlab.com, I found that this was possible using the uigetmodemanager function. The fix is a simple two lines:
hManager = uigetmodemanager(yourFigure);
hManager.CurrentMode.ModeStateData.textState = 0;