How to make the border trim the child elements?

You may write a converter for the Clip property. Converter should implement IMultiValueConverter and bound to actual size and corner radius, for example.

public class BorderClipConverter : IMultiValueConverter
{
    public object Convert(object[] values, Type targetType, object parameter, CultureInfo culture)
    {
        if (values.Length == 3 && values[0] is double && values[1] is double && values[2] is CornerRadius)
        {
            var width = (double)values[0];
            var height = (double)values[1];

            if (width < Double.Epsilon || height < Double.Epsilon)
            {
                return Geometry.Empty;
            }

            var radius = (CornerRadius)values[2];

            // Actually we need more complex geometry, when CornerRadius has different values.
            // But let me not to take this into account, and simplify example for a common value.
            var clip = new RectangleGeometry(new Rect(0, 0, width, height), radius.TopLeft, radius.TopLeft);
            clip.Freeze();

            return clip;
        }

        return DependencyProperty.UnsetValue;
    }

    public object[] ConvertBack(object value, Type[] targetTypes, object parameter, CultureInfo culture)
    {
        throw new NotSupportedException();
    }
}

Usage:

<Border CornerRadius="10">
    <Border.Clip>
        <MultiBinding Converter="{StaticResource BorderClipConverter}">
            <Binding Path="ActualWidth"
                        RelativeSource="{RelativeSource Self}"/>
            <Binding Path="ActualHeight"
                        RelativeSource="{RelativeSource Self}"/>
            <Binding Path="CornerRadius"
                        RelativeSource="{RelativeSource Self}"/>
        </MultiBinding>
    </Border.Clip>
</Border>