Dermot,
The document variable was a dead end and it is part of the Word document (Document.Variables). I didn't find anything useful that suppresses the prompt screen.
In Excel, you can link prompt values to cells, change the value in these cells by controlling Excel itself, and refresh your document with the Liveobjects.Refresh method.
However, I was stuck with Word, as there's no way to avoid the prompt windows to appear. So I ended up with a code that started another thread before calling the Refresh method, waited in the background for the LiveOffice window, then with SendKey, it filled the prompt and started the actual refresh.
I admit, it is not nice, but at least works.
If anyone from SAP looks at this post, please provide a better solution 
Greetings,
Marton