It turns out that Postman has very simple support for this, but the option I needed to make this work was just in a hidden column in the multipart form-data page.
What worked for me:
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Expose the “Content Type” Column in form-data and set to application/json
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Change the “file” option next to the KEY to “text” and then paste the raw JSON in the VALUE
…Of course it would be as simple as exposing a hidden column 