I wonder if something like this will work:
Add a reference to the System namespace in the declaration of each Window or UserControl where you want this.
xmlns:sys="clr-namespace:System;assembly=mscorlib"
Then in your resources section set things up like this:
<Window.Resources>
<ResourceDictionary>
<ResourceDictionary.MergedDictionaries>
<ResourceDictionary Source="styles.xaml"/>
</ResourceDictionary.MergedDictionaries>
<sys:String x:Key="WinConvertParam">IDTextForThisWindow</sys:String>
</ResourceDictionary>
</Window.Resources>
Your binding syntax could then look something like this:
{Binding SomeProperty,
Converter={StaticResource thatConverterIWrote},
ConverterParameter={StaticResource WinConvertParam}}
...and your Convert or ConvertBack methods in your conversion classes then become aware of the Window that's using them, provided you vary the value of that <sys:String/>
from file to file.
What do you think?