### 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_
"Error while invoking integration for ApiGateway ID %s: %s",
            invocation_context.api_id,
            e,
            exc_info=LOG.isEnabledFor(logging.DEBUG),
        )
        return e.to_response()
    except Exception as e:
        msg = f"Error invoking integration for API Gateway ID '{invocation_context.api_id}': {e}"
        LOG.exception(msg)
        return make_error_response(msg, 400)
def invoke_rest_api_integration_backend(invocation_context: ApiInvocationContext):
    
    invocation_path = invocation_context.path_with_query_string
    method = invocation_context.method
    headers = invocation_context.headers
    resource_path = invocation_context.resource_path
    integration = invocation_context.integration
    
    relative_path, query_string_params = extract_query_string_params(path=invocation_path)
    integration_type_orig = integration.get("type") or integration.get("integrationType") or ""
    integration_type = integration_type_orig.upper()
    integration_method = integration.get("httpMethod")
    uri = integration.get("uri") or integration.get("integrationUri") or ""
    try:
        invocation_context.path_params = extract_path_params(
            path=relative_path, extracted_path=resource_path
        )
    except Exception:
        invocation_context.path_params = {}
    if (uri.startswith("arn:aws:apigateway:") and ":lambda:path" in uri) or uri.startswith(