Stephen was already providing you the required Setting in the properties of the "Route via HTTP(S)"-Assertion.
Background is, that some HTTP-methods normally are NOT including any payload. And exactly for these methods the API-Gateway is ignoring such a payload by default.
The following is stated in docops:
Select the Transmit body regardless of request method check box to include the request body with the outbound request, even if the HTTP request method is one that normally would not include a body (for example, GET, HEAD, DELETE, or OPTIONS).
We had this issue in the past as well and I was also not aware of this behavior of the API-Gateway.
Ciao Stefan