### 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_
helpers.set_api_id_stage_invocation_path(invocation_context)
    try:
        return invoke_rest_api(invocation_context)
    except AuthorizationError as e:
        api_id = invocation_context.api_id
        return make_error_response("Not authorized to invoke REST API %s: %s" % (api_id, e), 403)
def invoke_rest_api(invocation_context: ApiInvocationContext):
    invocation_path = invocation_context.path_with_query_string
    raw_path = invocation_context.path or invocation_path
    method = invocation_context.method
    headers = invocation_context.headers
    
    authorize_invocation(invocation_context)
    extracted_path, resource = helpers.get_target_resource_details(invocation_context)
    if not resource:
        return make_error_response("Unable to find path %s" % invocation_context.path, 404)
    
    validator = RequestValidator(invocation_context)
    if not validator.is_request_valid():
        return make_error_response("Invalid request body", 400)
    api_key_required = resource.get("resourceMethods", {}).get(method, {}).get("apiKeyRequired")
    if api_key_required and not is_api_key_valid(invocation_context):
        return make_error_response("Access denied - invalid API key", 403)
    resource_methods = resource.get("resourceMethods", {})