You could just hide the controls textbox (with a smaller width), expose you're own (optionally set the IsEnabled to false) and use an Element binding and Converter. If you're using MVVM, then set the DataContext to your ViewModel. I suppose another option would be to overwrite the DataTemplate to not include the text box and do the same idea.
<StackPanel Orientation="Horizontal" Height="22">
<TextBox x:Name="textBox2" Width="106" Text="{Binding ElementName=datePicker2, Path=SelectedDate, Mode=TwoWay, Converter={StaticResource internationalDateTimeFormatConverter}}" />
<controls:DatePicker x:Name="datePicker2" IsTabStop="False" SelectedDate="{Binding TargetDatePicker, Mode=TwoWay, NotifyOnValidationError=true, ValidatesOnExceptions=true}" Width="23" HorizontalAlignment="Left" />
</StackPanel>
UPDATE:
The TwoWay binding from the text box to the the date picker works well, but it doesn't update the ViewModel Property. So I'm going to set the IsEnabled=False and call it good.