Two options comes to my mind:
1. Add handler to RadMenu ItemClick event and call focus on RadMenu. This way you will force TextBox to lose focus and update its binding.
2. Add handler to RadMenu ItemClick event and get the TextBox the take the binding and manually call Update on the binding. You can take the Binding using GetBindingExpression(TextBox.TextProperty) then ParentBinding.
I hope that this will help you.
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