OK, I was able to fix it.
At first, I found this Article: http://james.newtonking.com/archive/2009/10/23/efficient-json-with-json-net-reducing-serialized-json-size.aspx
The Article describes the usage of DefaultValueHandling.Ignore. I had already tried that before, and it did not work, so I was not surprised when it did not at the second attempt.
Then I realized I was still using the R2 version of JSON.NET and updated to R3. After that everything is working fine now.
I hope this is useful to someone...