### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
__slots__ = ["vtl"]
    def __init__(self):
        self.vtl = ApiGatewayVtlTemplate()
    def render(self, api_context: ApiInvocationContext) -> Union[bytes, str]:
        pass
    def render_vtl(self, template, variables):
        return self.vtl.render_vtl(template, variables=variables)
    @staticmethod
    def build_variables_mapping(api_context: ApiInvocationContext):
        
        return {
            "context": api_context.context or {},
            "stage_variables": api_context.stage_variables or {},
            "input": {
                "body": api_context.data_as_string(),
                "params": {
                    "path": api_context.path_params,
                    "querystring": api_context.query_params(),
                    
                    
                    
                    "header": dict(api_context.headers),
                },
            },
        }
class RequestTemplates(Templates):