Since you already have a custom source handler, you might find it easiest to remove the fields there, since you can run whatever Java code you want.
An alternative would be to configure a bunch of Single Message Transforms (SMTs) to remove fields, but I expect that could get really verbose – or might be impossible, since you want to remove different fields from different types of records.
I’m supposed to override the handle method, right?
Yes, that would work. Or the buildValue method if you think that would be less work. Either way, you’ll probably have to copy some code from the base class, since there’s not a great extension point just for transforming the document content. (Maybe that will change in the future.)