### 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_
uri = (
            invocation_context.integration.get("uri")
            or invocation_context.integration.get("integrationUri")
            or ""
        )
        action = uri.split("/")[-1]
        if invocation_context.integration.get("IntegrationType") == "AWS_PROXY":
            payload = self._create_request_parameters(invocation_context)
        elif APPLICATION_JSON in invocation_context.integration.get("requestTemplates", {}):
            payload = self.request_templates.render(invocation_context)
            payload = json.loads(payload)
        else:
            payload = json.loads(invocation_context.data)
        client = aws_stack.connect_to_service("stepfunctions")
        if isinstance(payload.get("input"), dict):
            payload["input"] = json.dumps(payload["input"])
        # Hot fix since step functions local package responses: Unsupported Operation: 'StartSyncExecution'
        method_name = (
            camel_to_snake_case(action) if action != "StartSyncExecution" else "start_execution"
        )
        try:
            # call met
            method = getattr(client, method_name)
        except AttributeError:
            msg = f"Invalid step function action: {method_name}"
            LOG.error(msg)
            return StepFunctionIntegration._create_response(
                HTTPStatus.BAD_REQUEST.value,