### 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_
#
        
        #
        
        # 
        # A 
        # A pa
        # A query string – http://example.com/foo?q=${stageVariables.<variable_name>}
        render_vars = {"stageVariables": invocation_context.stage_variables}
        rendered_uri = VtlTemplate().render_vtl(uri, render_vars)
        uri = apply_request_parameters(
            rendered_uri,
            integration=integration,
            path_params=path_params,
            query_params=query_string_params,
        )
        result = requests.request(method=method, url=uri, data=payload, headers=headers)
        if not result.ok:
            LOG.debug(
                "Upstream response from <%s> %s returned with status code: %s",
                method,
                uri,
                result.status_code,
            )
        
        invocation_context.response = result
        response_templates = ResponseTemplates()
        response_templates.render(invocation_context)
        return invocation_context.response
class SQSIntegration(BackendIntegration):