### 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_
input_path = target.get("InputPath")
    if input_path:
        event = extract_jsonpath(event, input_path)
    return event
def process_events(event: Dict, targets: List[Dict]):
    for target in targets:
        arn = target["Arn"]
        changed_event = filter_event_with_target_input_path(target, event)
        if target.get("Input"):
            changed_event = json.loads(target.get("Input"))
        try:
            send_event_to_target(
                arn,
                changed_event,
                pick_attributes(target, ["$.SqsParameters", "$.KinesisParameters"]),
                role=target.get("RoleArn"),
                target=target,
                source_service=ServicePrincipal.events,
                source_arn=target.get("RuleArn"),
            )
        except Exception as e:
            LOG.info(f"Unable to send event notification {truncate(event)} to target {target}: {e}")
def get_event_bus_name(event_bus_name_or_arn: Optional[EventBusNameOrArn] = None) -> str: