### 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_
def invoke(self, invocation_context: ApiInvocationContext):
        invocation_path = invocation_context.path_with_query_string
        integration = invocation_context.integration
        path_params = invocation_context.path_params
        method = invocation_context.method
        headers = invocation_context.headers
        relative_path, query_string_params = extract_query_string_params(path=invocation_path)
        uri = integration.get("uri") or integration.get("integrationUri") or ""
        if ":servicediscovery:" in uri:
            
            client = aws_stack.connect_to_service("servicediscovery")
            service_id = uri.split("/")[-1]
            instances = client.list_instances(ServiceId=service_id)["Instances"]
            instance = (instances or [None])[0]
            if instance and instance.get("Id"):
                uri = "http://%s/%s" % (instance["Id"], invocation_path.lstrip("/"))
        
        invocation_context.context = helpers.get_event_request_context(invocation_context)
        invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)
        request_templates = RequestTemplates()
        payload = request_templates.render(invocation_context)
        if isinstance(payload, dict):
            payload = json.dumps(payload)