I've been investigating this and found that WPF doesn't even expose an IsVisible (or Visibility) property for Inlines.
So I think you should rely on the TextBlock visibility instead, is that an option for you?

<TextBlock x:Name="Text1"><Run Text="{Binding ElementName=Text1, Mode=OneWay, Path=IsVisible}"/></TextBlock>

Oh right, yeah that should actually work for our problem!

Thanks for the help

I'm closing this then, if you fing any trouble with the TextBlock approach please reopen it again.