This article assumes you already have a working integration between AWS API Gateway and Amazon EventBridge.
It seems like you’re really trying to avoid using another lambda to handle the payload transformation. Good choice*. The number of lambdas in a serverless infrastructure can quickly get out-of-hand and you should avoid creating unnecessary resources whenever it’s possible.
Amazon REST API Gateway offers a solution in the form of Mapping templates. The basic passthrough integration looks as follows:
"Entries" : [
"DetailType": "foo type",