How could I potentially export the actual binary file from a POST request?

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There are several ways. The first way, as @grahamb pointed out, is to export the http object. But this will result in the multipart object that you would have to post-process yourself to get the binary object out.

Another way is to use the export packet bytes option. In the POST request, expand the MIME object in the details pane. Then expand the part that contains the binary file. The line just below the Content-Type header will contain your object. Right-click on it and choose for export packet bytes.

I'm not entirely sure your file will show up here, but you can try from the menu, File -> Export Objects -> HTTP ... and then select the object. More info in the User Guide here.