Remember, styles are independent entries which can be applied to more than one object to get the same look and feel for both. You should not be using Binding to get some external data into styles.
Rather use ControlTemplate to apply it to child elements or even apply it directly using DataContext.
< ... AccessText="{Binding RelativeSource={RelativeSource .... DisplayName ... }}"
Something like this or directly
AccessText="{Binding DisplayName}"
Making Binding Mode=TwoWay means the data will be updated in Both direction.