The RibbonSplitButton can behave like a toggle button by setting its IsToggle property to True. When the button is hosted in ApplicationMenu its default template is changed with a special one that adjust the UI of the button to the ApplicationMenu design. However, the alternative template is missing the code that updates the UI of the button when the IsChecked property is True.

To work this around, you can edit the ControlTemplate of RadRibbonSplitButton. The template for the button when hosted in ApplicationMenu is using the following x:Key "ApplicationRibbonSplitButtonTemplate". You can apply the edited template through the Template property of RadRibbonSplitButton.