Can you setup an alternate POST transition just for the use of the API that uses an alternate SUBMIT transition into the target workflow?  The API's SUBMIT doesn't have any required fields and maybe goes into a special "holding state".  Then after the transition is complete on the POSTING item, run a transition on the POSTED item and set the required fields and have it end up in the same state as if the user had been the one who initiated the POST transition on the original item.

I read this question a little differently. I would answer like this: Yes, the TransitionItem web service will allow you to run any transition, including a post transition. Generally speaking, you do not need to pass any details when running a Post transition because the details are defined in the transition in Composer. You don't need to define or pass any other details to the web service call.

If the answers here don't help, please provide more details about your situation. Are you getting an error? Did you try a TransitionItem that failed? How did it fail? Was the new item not created, or was the new items missing some data that you expected to see? etc.  If you would prefer, you can always open a case with the support team too.

The solution of PM Thomson would work, but I don't want to change the workflows too much. The question comes from a real world problem, so I know that I can specify a post transition in the call, but I get an error about missing required fields that come from the submit transition in the target workflow. The fields specified in the TransitionItem call always relate to the original item, not to the posted item in the target workflow.

A really ugly option:

For each field in the "destination/target" item that needs to be initialized, create a hidden/dummy field in the table of the "source" item.  Map these fields in the "Post" transition / Post Options / Mapping section.  You'll need to initialize the hidden fields in source item. 

The downside is that your source item's primary table will now have a bunch of fields that serve no purpose except as initialization data for posting to another App.

Yet another option is to do the posting from a State or Transition form in the source item workflow using the "SubmitToProject" JSON API function.