Since your question is a bit vague, please forgive if this is largely based on personal opinion: There are, to my knowledge, no other approaches to the mutable property model. I would however argue that you don't want one:
First of, functional programming, at least from a puristic point of view, attempts to avoid side effects. User interfaces, however, are exclusively about causing side effects. There is a slight philosophical mismatch to begin there.
One of the main benefits of immutable data is that you don't have to deal with control structures to avoid concurrent modification. However, JavaFX's event queue implements a very strict single-threaded approach with an implicit control of read and write access. On the other hand, user interface components fit the idea of mutable objects better than most other fields of programming. The node structure is, after all, an inherent hierarchy of stateful components.
Considering this, I think trying to force a functional and immutable paradigm on JavaFX is not going to work out. Instead, I would recommend building a translation layer based on keypath selections - e.g. binding a Label to display an (immutable) Person's name to the Person, not the name property, and having a resolver handle the access to the name attribute. Basically, this would mean having a combination of Bindings#select and a JavaBeanStringProperty.