Try adding UpdateSourceTrigger=PropertyChanged to the binding of your ComboBox's selected item like so:
<DataTemplate x:Key="ComboBoxPackagingType">
<ComboBox SelectedItem="{Binding PackagingType.SelectedItem, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged}" ItemsSource="{Binding PackagingType.ItemsSource}"/>
</DataTemplate>
This worked for me.