I've been delaying implementing the lessons learned here, giving me plenty of time to think about the exact right way to do it. As other people said, having a clear separation where the backend objects have listeners for when their properties change is definitely the way to go. Not only will it resolve the specific issue I was asking about in this question, it is going to make a lot of other bad design smells in this code look better. There are actually a lot of different Backend classes (going by the generic class names I used in my example), each with their own corresponding Panel class. And there's even a couple of places where some things can be moved around to separate other pairs of classes into Backend/Panel pairs following the same pattern and reducing a lot of passing junk around as parameters.
The rest of this answer is going to get language specific, as I am using Java.
I've not worried a whole lot about "JavaBeans," but I have found that following basic JavaBean conventions has been very helpful for me in the past: basically, using standard getters and setters for properties. Turns out there's a JavaBean convention I was unaware of which is really going to help here: bound properties. Bound properties are properties available through standard getters and setters which fire PropertyChangeEvents when they change. [I don't know for sure, but the JavaBeans standard may specify that all properties are supposed to be "bound properties." Not relevant to me, at this point. Be aware also that "standard" getters and setters can be very non-standard through the use of BeanInfo classes to define a JavaBean's exact interface, but I never use that, either.] (The main other JavaBean convention that I choose to follow or not as appropriate in each situation is a no-argument constructor; I'm already following it in this project because each of these Backend objects has to be serializable.)
I've found this blog entry, which was very helpful in cluing me into the bound properties/PropertyChangeEvents issue and helping me construct a plan for how I'm going to rework this code.
Right now all of my backend objects inherit from a common class called Model, which provides a couple of things every backend in this system needs including serialization support. I'm going to create an additional class JavaBean as a superclass of Model which will provide the PropertyChangeEvent support that I need, inherited by every Model. I'll update the setters in each Model to fire a PropertyChangeEvent when called. I may also have JavaBean inherited by a couple of classes which aren't technically Models in the same sense as these but which could also benefit from having other classes registered as listeners for them. The JavaBean class may not fully implement the JavaBean spec; as I've said, there are several details I don't care about. But it's good enough for this project. It sounds like I could get all this by inheriting from java.awt.Component, but these aren't components in any sense that I can justify, so I don't want to do that. (I also don't know what overhead it might entail.)
Once every Model is a JavaBean, complete with PropertyChangeEvent support, I'll do a lot of code cleanup: Models that are currently keeping references to Panels will be updated and the Panels will register themselves as listeners. So much cleaner! The Model won't have to know (and shouldn't have known in the first place) what methods the Panel should call on itself when the property updates.