### 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: str, integration: Dict[str, Any], path_params: Dict[str, str], query_params: Dict[str, str]
):
    request_parameters = integration.get("requestParameters")
    uri = uri or integration.get("uri") or integration.get("integrationUri") or ""
    if request_parameters:
        for key in path_params:
            # check if 
            request_param_key = f"integration.request.path.{key}"
            request_param_value = f"method.request.path.{key}"
            if request_parameters.get(request_param_key) == request_param_value:
                uri = uri.replace(f"{{{key}}}", path_params[key])
    if integration.get("type") != "HTTP_PROXY" and request_parameters:
        for key in query_params.copy():
            request_query_key = f"integration.request.querystring.{key}"
            request_param_val = f"method.request.querystring.{key}"
            if request_parameters.get(request_query_key, None) != request_param_val:
                query_params.pop(key)
    return add_query_params_to_url(uri, query_params)